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 ⋮ 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.⚹◊What is the relationship between the graph of a function $f$ and the sign of its derivative $f'$ in terms of increase or decrease?⍭ Generate me a similar question◊

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