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