Mathematics

Problem

Consider the piecewise–defined function

f(x)=\begin{cases} x^{2}+2x+1, & -3\le x<0,\\ x^{2/3}+3, & 0\le x\le 4. \end{cases}

The domain for this problem is the closed interval

[-3,4]

. (a) Verify that

f

is continuous on

[-3,4]

. (b) Compute

f'(x)

on each subinterval where the derivative exists, and determine all interior points of

[-3,4]

at which

f'(x)=0

. (c) Identify every interior point of

[-3,4]

at which

f'(x)

does not exist, and explain the geometric reason for the nondifferentiability at each such point. (d) Create a complete list of

x

–values that must be tested to locate the absolute maximum and minimum values of

f

[-3,4]

. (e) Evaluate

f

at all points found in part (d) and determine the absolute maximum value and the absolute minimum value of

f

[-3,4]

, stating where each occurs.

Intuitive explanation

This problem asks you to work with a function that changes its rule at $x=0$ , so you need to treat each piece separately and then think about how they fit together. The key insight is recognizing that the first piece is a familiar parabola (hint: try factoring), while the second piece involves a fractional exponent that creates a different shape. To find the range, extreme values, or sketch the graph, you'll evaluate each piece on its own domain, check the boundary behavior at $x=0$ , and compare all critical points and endpoints.

Intuitive explanation

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