How to find where a function is increasing at the greatest rate











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Given the function $f(x) = frac{1000x^2}{11+x^2}$ on the interval $[0, 3]$, how would I calculate where the function is increasing at the greatest rate?




Differentiating the function will give its slope. Since slope is defined as the rate of change, then getting the maxima of the function's derivative will indicate where it is increasing at the greatest rate.



The derivative of $f(x)$ is $frac{22000x}{(11+x^2)^2}$



Applying the first derivative test, the critical number is $sqrt{frac{11}{3}}$. The function increases before the critical number and decreases after it, so the critical number is a maximum. $sqrt{frac{11}{3}}$ is the answer.






calculus functions proof-verification







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  • 1




    What have you tried? What do you know about calculating maxima? If you don't demonstrate any effort or context, we can't help you very effectively.
    – Zach Stone
    Dec 18 '15 at 4:30










  • "...so it is a maxima" - what is it in this statement? (And it should just be called a maximum, singular)
    – The Chaz 2.0
    Dec 18 '15 at 5:06










  • @The Chaz 2.0, Edited!
    – hmir
    Dec 18 '15 at 5:08








  • 2




    Right. "Where the function reaches its maximum on the interval" is not the same as "where the rate of change is maximized on the interval". You need to take the second derivative.
    – The Chaz 2.0
    Dec 18 '15 at 5:11










  • It is not without interest for the beginner to look at the following difference where the fastest rate of growth is noted shortly before $sqrt{frac{11}{3}}$: $$fleft(sqrt{dfrac{11}{3}+0.1}right)-fleft(sqrt{dfrac{11}{3}}right)approx5.07900\fleft(sqrt{dfrac{11}{3}}right)-fleft(sqrt{dfrac{11}{3}-0.1}right)approx5.14875$$
    – Piquito
    Apr 5 at 12:07















up vote
6
down vote

favorite
1












Given the function $f(x) = frac{1000x^2}{11+x^2}$ on the interval $[0, 3]$, how would I calculate where the function is increasing at the greatest rate?




Differentiating the function will give its slope. Since slope is defined as the rate of change, then getting the maxima of the function's derivative will indicate where it is increasing at the greatest rate.



The derivative of $f(x)$ is $frac{22000x}{(11+x^2)^2}$



Applying the first derivative test, the critical number is $sqrt{frac{11}{3}}$. The function increases before the critical number and decreases after it, so the critical number is a maximum. $sqrt{frac{11}{3}}$ is the answer.






calculus functions proof-verification







share|cite|improve this question




















  • 1




    What have you tried? What do you know about calculating maxima? If you don't demonstrate any effort or context, we can't help you very effectively.
    – Zach Stone
    Dec 18 '15 at 4:30










  • "...so it is a maxima" - what is it in this statement? (And it should just be called a maximum, singular)
    – The Chaz 2.0
    Dec 18 '15 at 5:06










  • @The Chaz 2.0, Edited!
    – hmir
    Dec 18 '15 at 5:08








  • 2




    Right. "Where the function reaches its maximum on the interval" is not the same as "where the rate of change is maximized on the interval". You need to take the second derivative.
    – The Chaz 2.0
    Dec 18 '15 at 5:11










  • It is not without interest for the beginner to look at the following difference where the fastest rate of growth is noted shortly before $sqrt{frac{11}{3}}$: $$fleft(sqrt{dfrac{11}{3}+0.1}right)-fleft(sqrt{dfrac{11}{3}}right)approx5.07900\fleft(sqrt{dfrac{11}{3}}right)-fleft(sqrt{dfrac{11}{3}-0.1}right)approx5.14875$$
    – Piquito
    Apr 5 at 12:07













up vote
6
down vote

favorite
1









up vote
6
down vote

favorite
1






1





Given the function $f(x) = frac{1000x^2}{11+x^2}$ on the interval $[0, 3]$, how would I calculate where the function is increasing at the greatest rate?




Differentiating the function will give its slope. Since slope is defined as the rate of change, then getting the maxima of the function's derivative will indicate where it is increasing at the greatest rate.



The derivative of $f(x)$ is $frac{22000x}{(11+x^2)^2}$



Applying the first derivative test, the critical number is $sqrt{frac{11}{3}}$. The function increases before the critical number and decreases after it, so the critical number is a maximum. $sqrt{frac{11}{3}}$ is the answer.






calculus functions proof-verification







share|cite|improve this question















Given the function $f(x) = frac{1000x^2}{11+x^2}$ on the interval $[0, 3]$, how would I calculate where the function is increasing at the greatest rate?




Differentiating the function will give its slope. Since slope is defined as the rate of change, then getting the maxima of the function's derivative will indicate where it is increasing at the greatest rate.



The derivative of $f(x)$ is $frac{22000x}{(11+x^2)^2}$



Applying the first derivative test, the critical number is $sqrt{frac{11}{3}}$. The function increases before the critical number and decreases after it, so the critical number is a maximum. $sqrt{frac{11}{3}}$ is the answer.






calculus functions proof-verification




calculus functions proof-verification






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 18 '15 at 5:08

























asked Dec 18 '15 at 4:27









hmir

16016




16016








  • 1




    What have you tried? What do you know about calculating maxima? If you don't demonstrate any effort or context, we can't help you very effectively.
    – Zach Stone
    Dec 18 '15 at 4:30










  • "...so it is a maxima" - what is it in this statement? (And it should just be called a maximum, singular)
    – The Chaz 2.0
    Dec 18 '15 at 5:06










  • @The Chaz 2.0, Edited!
    – hmir
    Dec 18 '15 at 5:08








  • 2




    Right. "Where the function reaches its maximum on the interval" is not the same as "where the rate of change is maximized on the interval". You need to take the second derivative.
    – The Chaz 2.0
    Dec 18 '15 at 5:11










  • It is not without interest for the beginner to look at the following difference where the fastest rate of growth is noted shortly before $sqrt{frac{11}{3}}$: $$fleft(sqrt{dfrac{11}{3}+0.1}right)-fleft(sqrt{dfrac{11}{3}}right)approx5.07900\fleft(sqrt{dfrac{11}{3}}right)-fleft(sqrt{dfrac{11}{3}-0.1}right)approx5.14875$$
    – Piquito
    Apr 5 at 12:07














  • 1




    What have you tried? What do you know about calculating maxima? If you don't demonstrate any effort or context, we can't help you very effectively.
    – Zach Stone
    Dec 18 '15 at 4:30










  • "...so it is a maxima" - what is it in this statement? (And it should just be called a maximum, singular)
    – The Chaz 2.0
    Dec 18 '15 at 5:06










  • @The Chaz 2.0, Edited!
    – hmir
    Dec 18 '15 at 5:08








  • 2




    Right. "Where the function reaches its maximum on the interval" is not the same as "where the rate of change is maximized on the interval". You need to take the second derivative.
    – The Chaz 2.0
    Dec 18 '15 at 5:11










  • It is not without interest for the beginner to look at the following difference where the fastest rate of growth is noted shortly before $sqrt{frac{11}{3}}$: $$fleft(sqrt{dfrac{11}{3}+0.1}right)-fleft(sqrt{dfrac{11}{3}}right)approx5.07900\fleft(sqrt{dfrac{11}{3}}right)-fleft(sqrt{dfrac{11}{3}-0.1}right)approx5.14875$$
    – Piquito
    Apr 5 at 12:07








1




1




What have you tried? What do you know about calculating maxima? If you don't demonstrate any effort or context, we can't help you very effectively.
– Zach Stone
Dec 18 '15 at 4:30




What have you tried? What do you know about calculating maxima? If you don't demonstrate any effort or context, we can't help you very effectively.
– Zach Stone
Dec 18 '15 at 4:30












"...so it is a maxima" - what is it in this statement? (And it should just be called a maximum, singular)
– The Chaz 2.0
Dec 18 '15 at 5:06




"...so it is a maxima" - what is it in this statement? (And it should just be called a maximum, singular)
– The Chaz 2.0
Dec 18 '15 at 5:06












@The Chaz 2.0, Edited!
– hmir
Dec 18 '15 at 5:08






@The Chaz 2.0, Edited!
– hmir
Dec 18 '15 at 5:08






2




2




Right. "Where the function reaches its maximum on the interval" is not the same as "where the rate of change is maximized on the interval". You need to take the second derivative.
– The Chaz 2.0
Dec 18 '15 at 5:11




Right. "Where the function reaches its maximum on the interval" is not the same as "where the rate of change is maximized on the interval". You need to take the second derivative.
– The Chaz 2.0
Dec 18 '15 at 5:11












It is not without interest for the beginner to look at the following difference where the fastest rate of growth is noted shortly before $sqrt{frac{11}{3}}$: $$fleft(sqrt{dfrac{11}{3}+0.1}right)-fleft(sqrt{dfrac{11}{3}}right)approx5.07900\fleft(sqrt{dfrac{11}{3}}right)-fleft(sqrt{dfrac{11}{3}-0.1}right)approx5.14875$$
– Piquito
Apr 5 at 12:07




It is not without interest for the beginner to look at the following difference where the fastest rate of growth is noted shortly before $sqrt{frac{11}{3}}$: $$fleft(sqrt{dfrac{11}{3}+0.1}right)-fleft(sqrt{dfrac{11}{3}}right)approx5.07900\fleft(sqrt{dfrac{11}{3}}right)-fleft(sqrt{dfrac{11}{3}-0.1}right)approx5.14875$$
– Piquito
Apr 5 at 12:07










1 Answer
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Differentiating the function will give its slope. Since slope is defined as the rate of change, then getting the maxima of the function's derivative will indicate where it is increasing at the greatest rate.



The derivative of $f(x)$ is $frac{22000x}{(11+x^2)^2}$



Applying the first derivative test, the critical number is $sqrt{frac{11}{3}}$. The function increases before the critical number and decreases after it, so the critical number is a maximum. $sqrt{frac{11}{3}}$ is the answer.






share|cite|improve this answer



















  • 1




    This should go up above! I copied this answer and put it in the body of your question.
    – graydad
    Dec 18 '15 at 5:04











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up vote
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Differentiating the function will give its slope. Since slope is defined as the rate of change, then getting the maxima of the function's derivative will indicate where it is increasing at the greatest rate.



The derivative of $f(x)$ is $frac{22000x}{(11+x^2)^2}$



Applying the first derivative test, the critical number is $sqrt{frac{11}{3}}$. The function increases before the critical number and decreases after it, so the critical number is a maximum. $sqrt{frac{11}{3}}$ is the answer.






share|cite|improve this answer



















  • 1




    This should go up above! I copied this answer and put it in the body of your question.
    – graydad
    Dec 18 '15 at 5:04















up vote
0
down vote













Differentiating the function will give its slope. Since slope is defined as the rate of change, then getting the maxima of the function's derivative will indicate where it is increasing at the greatest rate.



The derivative of $f(x)$ is $frac{22000x}{(11+x^2)^2}$



Applying the first derivative test, the critical number is $sqrt{frac{11}{3}}$. The function increases before the critical number and decreases after it, so the critical number is a maximum. $sqrt{frac{11}{3}}$ is the answer.






share|cite|improve this answer



















  • 1




    This should go up above! I copied this answer and put it in the body of your question.
    – graydad
    Dec 18 '15 at 5:04













up vote
0
down vote










up vote
0
down vote









Differentiating the function will give its slope. Since slope is defined as the rate of change, then getting the maxima of the function's derivative will indicate where it is increasing at the greatest rate.



The derivative of $f(x)$ is $frac{22000x}{(11+x^2)^2}$



Applying the first derivative test, the critical number is $sqrt{frac{11}{3}}$. The function increases before the critical number and decreases after it, so the critical number is a maximum. $sqrt{frac{11}{3}}$ is the answer.






share|cite|improve this answer














Differentiating the function will give its slope. Since slope is defined as the rate of change, then getting the maxima of the function's derivative will indicate where it is increasing at the greatest rate.



The derivative of $f(x)$ is $frac{22000x}{(11+x^2)^2}$



Applying the first derivative test, the critical number is $sqrt{frac{11}{3}}$. The function increases before the critical number and decreases after it, so the critical number is a maximum. $sqrt{frac{11}{3}}$ is the answer.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 18 '15 at 5:08

























answered Dec 18 '15 at 4:48









hmir

16016




16016








  • 1




    This should go up above! I copied this answer and put it in the body of your question.
    – graydad
    Dec 18 '15 at 5:04














  • 1




    This should go up above! I copied this answer and put it in the body of your question.
    – graydad
    Dec 18 '15 at 5:04








1




1




This should go up above! I copied this answer and put it in the body of your question.
– graydad
Dec 18 '15 at 5:04




This should go up above! I copied this answer and put it in the body of your question.
– graydad
Dec 18 '15 at 5:04


















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