Question

For what values of the numbers a and b does the function below have maximum value f(1) = 4? (Round the answers to three decimal places.). f(x) = axe^[b(x^2)].

Answer 1

f (x) = $a x e^{b x^{2} }$
f ` (x) = $a( e^{b x^{2} } +x e ^{b x^{2} } *2 b x )=a e ^{b x^{2} } (1+2b x^{2} )$
f ( 1 ) = 4
f ` ( 1 ) = 0
$4 = a e ^{b} \\ a e ^{b} (1 + 2 b ) = 0 \\ 4 ( 1 + 2 b )= 0$
b = -1/2
$4=a e ^{-1/2} = \frac{a}{ \sqrt{e} } \\ a = 4 \sqrt{e}$
Answer:  a = 4√e, b = -1/2