Question

Math

Posted 4 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 4 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

From the graph, we observe that the line representing $f'$ is horizontal and below the x-axis, indicating that $f'$ is negative

step 3

Since $f'$ is negative, it means that the slope of the tangent line to the graph of $f$ is negative, which implies that $f$ is decreasing

step 4

Therefore, the appropriate calculus-based justification for the fact that $f$ is decreasing is that $f'$ is negative

Answer

(C) $f'$ is negative.

Key Concept

Derivative and Function Behavior

Explanation

The derivative $f'$ indicates the slope of the function $f$. If $f'$ is negative, it means the function $f$ is decreasing.

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