Question

Math

Posted 6 months ago

```
This is the graph of $f$.
Let $g(x)=\int_{0}^{x} f(t) d t$.
What is an appropriate calculus-based justification for the fact that $g$ is concave up?
Choose 1 answer:
(A) $f$ is positive.
(B) $f$ is increasing.
(C) $f$ is concave up.
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

We are given the function $g(x) = \int_{0}^{x} f(t) \, dt$ and need to determine why $g$ is concave up

step 2

To determine the concavity of $g(x)$, we need to consider the second derivative of $g(x)$

step 3

By the Fundamental Theorem of Calculus, the first derivative of $g(x)$ is $g'(x) = f(x)$

step 4

The second derivative of $g(x)$ is $g''(x) = f'(x)$

step 5

For $g(x)$ to be concave up, $g''(x)$ must be positive. This means $f'(x) > 0$

step 6

Since the graph of $f$ is increasing, $f'(x) > 0$. Therefore, $g(x)$ is concave up because $f$ is increasing

Answer

(B) $f$ is increasing.

Key Concept

Concavity of an integral function

Explanation

The concavity of the integral function $g(x)$ is determined by the sign of the second derivative $g''(x)$. Since $g''(x) = f'(x)$ and $f$ is increasing, $f'(x) > 0$, making $g(x)$ concave up.

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