Exer. 79-82: (a) Sketch the graph of f on the given interval...

Exer. $79-82:$ (a) Sketch the graph of $f$ on the given interval $[a, b] .$ (b) Estimate the range of $f$ on $[a, b] .$ (c) Estimate the intervals on which $f$ is increasing or is decreasing.
$$f(x)=x^{5}-3 x^{2}+1 ; \quad[-0.7,1.4]$$

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Key Concepts

Graphing Functions

Graphing functions involves representing a function visually on a coordinate system over a specified interval. This process includes plotting key points such as intercepts, end points, and critical points, and using information like symmetry and asymptotic behavior to create an accurate sketch of the function's behavior.

Function Range

The range of a function on an interval is the collection of all output values the function takes as the input varies over that interval. Determining the range often involves evaluating the function at endpoints and at critical points where local maximums and minimums occur, ensuring a complete understanding of the function's output behavior.

Monotonicity and Increasing/Decreasing Intervals

Monotonicity refers to the property of a function being exclusively increasing or decreasing over certain intervals. By analyzing the first derivative of a function, one can identify intervals where the function is rising (increasing) or falling (decreasing). This analysis is critical in understanding the overall behavior, trends, and local extrema of the function on a given interval.

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