Mathematics

Problem

Let the function

f

be defined on the interval

-2\le x\le 6

f(x)=\begin{cases}(x+2)^2,& -2\le x\le 2,\\6x-x^{2},& 2<x\le6.\end{cases}

\nDefine the accumulation function

g(x)=\int_{0}^{x} f(t)\,dt,

\nfor all

x

[-2,6]

. (a) Compute

f'(x)

on each sub-interval of the domain. (b) Determine the sign of

f'(x)

on the intervals

(-2,2)

and

(2,6)

. Provide supporting work. (c) Using your conclusions in part (b), indicate the intervals on which

g

is concave up and the intervals on which

g

is concave down. (d) Identify all

x

-coordinates in the open interval

(-2,6)

at which the graph of

g

could have a point of inflection. For each such

x

-value, justify whether a point of inflection actually occurs.

Intuitive explanation

The key to understanding $g(x)$ is recognizing it as a cumulative sum of the area under $f$ —as $x$ increases, $g(x)$ grows by adding more area beneath the curve. Since $f$ is defined piecewise with different formulas on different intervals, you'll need to split your integration at $x=2$ and handle each piece separately. The critical insight is that $g(x)$ inherits information about $f$ 's behavior: where $f$ is positive, $g$ increases; where $f$ is negative, $g$ decreases; and where $f$ has a maximum or minimum, $g$ has a zero slope.

Intuitive explanation

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