Maxima of sum of two gaussians
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I am trying to find an analytical form of the maxima of the function
$$f(x) = a_1 e^{-b^2_1 x^2} + a_2 e^{-b^2_2 (x+x_c)^2} , tag{1}$$
such that I can define a function $g(x)$ that has the maximum value $max (g(x)) = 1$, i.e.
$$g(x):= frac{f(x)}{max (f(x))} $$
The parameter $x_c$ is such that the maxima is unique or in other words only one maxima exists. For example, take the case where $x_c = 0.017,b^2_1=10,b^2_2=3,a_1=0.8,a_2=0.2$. The function $f(x)$ has only a single maxima with these parameters.
I could only come up with an approximate solution where $max(g(x)) approx 1$, i.e.
$$g(x):= frac{f(x)}{a_1 + a_2 e^{-b^2_2 (x_c)^2}} $$
Is there an exact solution to my problem?.
calculus real-analysis optimization
asked Nov 15 at 23:16
jsid
113
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up vote
2
down vote
favorite
I am trying to find an analytical form of the maxima of the function
$$f(x) = a_1 e^{-b^2_1 x^2} + a_2 e^{-b^2_2 (x+x_c)^2} , tag{1}$$
such that I can define a function $g(x)$ that has the maximum value $max (g(x)) = 1$, i.e.
$$g(x):= frac{f(x)}{max (f(x))} $$
The parameter $x_c$ is such that the maxima is unique or in other words only one maxima exists. For example, take the case where $x_c = 0.017,b^2_1=10,b^2_2=3,a_1=0.8,a_2=0.2$. The function $f(x)$ has only a single maxima with these parameters.
I could only come up with an approximate solution where $max(g(x)) approx 1$, i.e.
$$g(x):= frac{f(x)}{a_1 + a_2 e^{-b^2_2 (x_c)^2}} $$
Is there an exact solution to my problem?.
calculus real-analysis optimization
asked Nov 15 at 23:16
jsid
113
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am trying to find an analytical form of the maxima of the function
$$f(x) = a_1 e^{-b^2_1 x^2} + a_2 e^{-b^2_2 (x+x_c)^2} , tag{1}$$
such that I can define a function $g(x)$ that has the maximum value $max (g(x)) = 1$, i.e.
$$g(x):= frac{f(x)}{max (f(x))} $$
The parameter $x_c$ is such that the maxima is unique or in other words only one maxima exists. For example, take the case where $x_c = 0.017,b^2_1=10,b^2_2=3,a_1=0.8,a_2=0.2$. The function $f(x)$ has only a single maxima with these parameters.
I could only come up with an approximate solution where $max(g(x)) approx 1$, i.e.
$$g(x):= frac{f(x)}{a_1 + a_2 e^{-b^2_2 (x_c)^2}} $$
Is there an exact solution to my problem?.
calculus real-analysis optimization
asked Nov 15 at 23:16
jsid
113
I am trying to find an analytical form of the maxima of the function
$$f(x) = a_1 e^{-b^2_1 x^2} + a_2 e^{-b^2_2 (x+x_c)^2} , tag{1}$$
such that I can define a function $g(x)$ that has the maximum value $max (g(x)) = 1$, i.e.
$$g(x):= frac{f(x)}{max (f(x))} $$
The parameter $x_c$ is such that the maxima is unique or in other words only one maxima exists. For example, take the case where $x_c = 0.017,b^2_1=10,b^2_2=3,a_1=0.8,a_2=0.2$. The function $f(x)$ has only a single maxima with these parameters.
I could only come up with an approximate solution where $max(g(x)) approx 1$, i.e.
$$g(x):= frac{f(x)}{a_1 + a_2 e^{-b^2_2 (x_c)^2}} $$
Is there an exact solution to my problem?.
calculus real-analysis optimization
calculus real-analysis optimization
asked Nov 15 at 23:16
jsid
113
asked Nov 15 at 23:16
jsid
113
asked Nov 15 at 23:16
jsid
113
asked Nov 15 at 23:16
jsid
113
asked Nov 15 at 23:16
jsid
113
113
add a comment |
add a comment |
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