MATH SOLVE

5 months ago

Q:
# Find constants a and b in the function f(x)=axe^bx such that f(1/5)=1 and the function has a local maximum at x=1/5.

Accepted Solution

A:

F (x) = axe ^ bx
F β (x) = d/dx (axe ^ bx)
= a (d/dx) (xe ^ bx)
= a [(d/dx * x) e ^ bx + (x * d/dx * e ^ bx)]
= a [e ^ bx + bxe ^ bx]
= a [1 + bx] e ^ bx
Β
For critical points, set f β (x) = 0 and solve for x
Β
a [1 + bx] e ^ bx = 0
= 1 + bx = 0
= bx = -1
X = -1/b
Given that f (x) has local max at 1/5, critical point x =
1/5
-1/5 = -1/b
Therefore b = -5
Β
F(x) = axe ^ -5x
F(1/5) = a/5 e ^-1
1 = a/5e
Β
Therefore, a = 5e