Find the critical numbers of f (if any). Find the open...

Find the critical numbers of $f$ (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results.
$$
f(x)=2^{x^{2}-3}
$$

Key Concepts

Graphing Utility

A graphing utility is a technological tool used to visualize functions and their properties such as increasing/decreasing intervals and relative extrema. It helps confirm analytic results by providing a graphical representation, which can be crucial for understanding and verifying the behavior of more complex functions.

Critical Numbers

Critical numbers are the values in the domain of the function at which the derivative is zero or undefined. They are important because they are the potential locations for local maxima, minima, or points of inflection, and play a key role in the analysis of the function's behavior.

Increasing/Decreasing Intervals

The open intervals on which a function is increasing or decreasing are determined by the sign of its derivative. When the derivative is positive over an interval, the function is increasing; when it is negative, the function is decreasing. This information helps in understanding the overall behavior and trend of the function.

Relative Extrema

Relative extrema refer to the local maximum and minimum values of a function. These are found at critical numbers where the derivative changes sign. Identifying these points helps in understanding the peaks and valleys in the graph of the function.

Chain Rule

The Chain Rule is a fundamental technique in calculus for differentiating composite functions. It is used when a function is composed of another function, and it allows the derivative of the composite function to be expressed as the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function.

Exponential Functions

Exponential functions are functions in which the variable appears in the exponent, and they often involve a constant base raised to a variable exponent. Their differentiation usually involves the natural logarithm of the base and the application of the chain rule when the exponent is a function of the variable.

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