a. Suppose that the function y=f^'(x) is decreasing on the interval (0,2) and f^'(1)=0 . What ca

step 1 The following problem, we want to let F be a function with F of X greater than 0 for all X. Then

A. Suppose that the function y=f^'(x) is decreasing on the...

a. Suppose that the function $y=f^{\prime}(x)$ is decreasing on the interval (0,2) and $f^{\prime}(1)=0 .$ What can you say about a possible relative extremum of $f(x)$ at $x=1 ?$ Justify your answer. b. Suppose that the function $y=g^{\prime}(x)$ is increasing on the interval (0,2) and $g^{\prime}(1)=0 .$ What can you say about a possible relative extremum of $g(x)$ at $x=1$ ? Justify your answer.

a. Suppose that the function $y=f^{\prime}(x)$ is decreasing on the interval (0,2) and $f^{\prime}(1)=0 .$ What can you say about a possible relative extremum of $f(x)$ at $x=1 ?$ Justify your answer.
b. Suppose that the function $y=g^{\prime}(x)$ is increasing on the interval (0,2) and $g^{\prime}(1)=0 .$ What can you say about a possible relative extremum of $g(x)$ at $x=1$ ? Justify your answer.

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Key Concepts

Critical Points

A critical point is a value in the domain of a function where its derivative equals zero or does not exist. Critical points are important because they are candidates for relative extrema, where the function may have a local maximum or minimum.

Monotonicity of the Derivative

The behavior of the derivative—in particular, whether it is increasing or decreasing—is crucial in understanding how the function behaves around a critical point. When the derivative is monotonic, its sign before and after the zero can be determined more confidently, which in turn informs us about the function’s local behavior.

First Derivative Test

This test uses the sign change of the derivative around a critical point to determine the nature of a relative extremum. If the derivative changes from positive to negative at a critical point, the function has a relative maximum; if it changes from negative to positive, the function has a relative minimum. The monotonic behavior of the derivative, such as being increasing or decreasing, helps in applying this test by clearly indicating the trend of the derivative's values on either side of the critical point.

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