Find all critical values, the largest open intervals on...

Find all critical values, the largest open intervals on which $f$ is increasing, the largest open intervals on which $f$ is decreasing, and all relative maxima and minima. Sketch a rough graph of $f$. In Exercises 37 through 42, assume that the constants $a$ and $b$ are positive. $f(x)=5+20 x-x^{5}$

Key Concepts

Critical Points

Critical points are values of the variable where the derivative of the function is zero or undefined. They indicate potential points of local extremum, which need further analysis to determine whether the function has a local maximum, minimum, or neither at those points.

First Derivative Test

The first derivative test involves analyzing the sign of the derivative before and after a critical point. This test helps determine if the function is increasing or decreasing around that point, and thus whether a critical value corresponds to a local maximum or minimum.

Increasing/Decreasing Intervals

Increasing intervals are sections of the domain where the derivative of the function is positive, while decreasing intervals are sections where the derivative is negative. Identifying these intervals provides insight into the overall behavior and monotonicity of the function.

Relative Extrema

Relative extrema are the local maximum and minimum values that a function attains. They occur at critical points where the function changes its increasing/decreasing behavior. Determining these points is essential for understanding the function’s overall structure and for graphing.

Graphical Analysis of Functions

Graphical analysis involves sketching the function using critical points, increasing/decreasing intervals, and end behavior. For polynomial functions, it is important to consider the degree and leading coefficient to deduce the global behavior, which aids in constructing an accurate rough graph of the function.

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