philosphy111of
2021-11-18
Answered

The graph of a function f is shown. Which graph is an antiderivative of f?

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Coldst

Answered 2021-11-19
Author has **18** answers

Step 1

Let$g\left(x\right)$ be the antiderivative of $f\left(x\right)$ . Then we must have

$g\left(x\right)=\int f\left(x\right)dx$

$\Rightarrow {g}^{\prime}\left(x\right)=f\left(x\right)$

Now, since$f\left(x\right)$ is negative $(-\mathrm{\infty},\text{}p)$ that means the ${g}^{\prime}\left(x\right)$ is negative in this interval. So, g(x) (the antiderivative of $f\left(x\right)$ must be decreasing in this interval. This scenario is shown in two graphs a and b.

Again we notice that$f\left(x\right)$ is positive that means, ${g}^{\prime}\left(x\right)$ is positive in $(p,\text{}\mathrm{\infty})$ . So, the function $g\left(x\right)=$ antiderivative of $f\left(x\right)$ must be increasing in this interval and from the graph we can see that the graph b is increasing after the point $x=p$ .

So, the correct graph for the antiderivative of$f\left(x\right)$ is b.

Let

Now, since

Again we notice that

So, the correct graph for the antiderivative of

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