Question

Math

Posted 5 months ago

```
The differentiable function $f$ and its derivative $f^{\prime}$ are graphed.
What is an appropriate calculus-based justification for the fact that $f$ is decreasing?
Choose 1 answer:
A) $f^{\prime}$ is constant.
(B) As the $x$-values increase, the values of $f$ decrease.
(C) $f^{\prime}$ is negative.
```

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

To determine why the function $f$ is decreasing, we need to analyze the behavior of its derivative $f'$

step 2

The graph shows that $f'$ is a horizontal line, which means $f'$ is constant

step 3

Since $f'$ is a horizontal line below the x-axis, it indicates that $f'$ is negative

step 4

A negative derivative $f'$ implies that the function $f$ is decreasing

Answer

(C) $f^{\prime}$ is negative.

Key Concept

Derivative and Function Behavior

Explanation

A function $f$ is decreasing if its derivative $f'$ is negative. In this case, the graph shows that $f'$ is a constant negative value, which justifies that $f$ is decreasing.

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