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.

Question

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.

Accepted Answer

∻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.⚹◊What would be the mathematical expression for the derivative $f'(x)$ of the function $f(x)$ in this scenario?⍭ Generate me a similar question◊

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