Thread: functions and derivatives

Hi !

I need some help solving a function. I've attached the problem.

Thank you !

Originally Posted by Vilkku

Hi !

I need some help solving a function. I've attached the problem.
Thank you !

Hello, Vilkku,

I'm refering to the function f(x) = 2x*e^x

a) Since e^x > 0 for all x f(x) = 0 only if x = 0
A produkt of 2 factors is positive (that means greater than zero) if both factors have the same sign. Since e^x > 0 for all x f(x) > 0 if x > 0.
f(x) < 0 if x < 0

b) Use product rule:
f'(x) = e^x * 2 + 2x * e^x = e^x(2 + 2x)
f is increasing if f'(x) > 0. According to the considerations of a)
f is increasing if x > -1
f is decreasing if -∞ < x < -1
Thus at x = -1 is a (absolute) minimum

c) Use product rule:
f''(x) = (2+2x)*e^x + e^x * 2 = e^x(4+2x)
f is convex if f''(x) > 0
According to the considerations of a)
f''(x) > 0 if x > -2
f is concave if f''(x) < 0. Thus
f''(x) < 0 if -∞ < x < -2
Therefore the function has a point of inflection at (-2, -4/e^2)

I've added a sketch of the graph

EB