Let the function be defined piecewise by For ,

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let the function

f

be defined piecewise by For

x \leq 0

f(x) = x^3 - 3x^2

. For

x > 0

f(x) = 2x^3 - 6x + 1

. The domain of

f

is all real numbers. (a) Verify that

f

is differentiable at

x = 0

. (b) Find

f''(x)

on each interval

x < 0

and

x > 0

. (c) Determine all intervals on which the graph of

f

is concave up and all intervals on which it is concave down. (d) Identify every point of inflection of

f

, and justify each conclusion.