for what value of the numbers a and b does the function below have maximum value f(2)=4 f(x)=axe^(bx^2) also f'(x) and f(x) must equal for when the max occurrs at x=?x

Question

for what value of the numbers a and b does the function below have maximum value f(2)=4  f(x)=axe^(bx^2)  also f'(x) and f(x) must equal for when the max occurrs at x=?x

anonymous · Answer

First find the derivative:
$$f'(x)=ae^{bx^2}+2abx~e^{bx^2}$$
Find the critical points:
$$ae^{bx^2}+2abx~e^{bx^2}=0~~\iff~~1+2bx=0$$
You can find (x\) in terms of $b$.

Knowing that a max is supposed to occur when $x=2$, you will be able to find what value $b$ would take on.

To find $a$, you would substitute your solution for $b$ and the given value for $f(2)$, then solve the resulting equation for $a$.
$$f(2)=4=2ae^{4b}$$