Mathematics

Problem

Consider a twice–differentiable function

f

whose derivative is given by

f'(x)=(x-1)^2(x+2)

. Which of the following statements must be true? a)

f

has a relative maximum at

x=-2

and points of inflection at

x=-1

and

x=1

. b)

f

has no relative extrema because

f'(x)

is always nonnegative. c)

f

has a relative minimum at

x=-2

and points of inflection at

x=-1

and

x=1

. d)

f

has a relative maximum at

x=1

and is concave down for

x>1

Intuitive explanation

The key is to recognize that $f'(x)$ tells you where $f$ is increasing or decreasing, and where its concavity changes. Notice that $f'(x)$ has a double root at $x=1$ and a simple root at $x=-2$ —this means $f$ touches the $x$ -axis at $x=1$ without crossing it (so $x=1$ is neither a local max nor min), while $f'$ actually changes sign at $x=-2$ . Use a sign chart on $f'(x)$ to determine monotonicity, and then analyze $f''(x)$ to find inflection points where concavity flips.

Intuitive explanation

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