This is not a recipe for you simply to follow, but rather a series of steps to help guide your thinking that will often work well.

Compute the function’s derivative, $f'(x)$.

Find all critical numbers c such that either:

the derivative is zero: $f'(c) = 0$;

or the derivative $f'(c)$ does not exist.

Make a list of the values c that you find. (If you are examining a closed interval ($a \le x \le b$), then you may be able to skip to step 4.)

For each critical point c, check whether $f$ is a local maximum or minimum or neither. Use either the First Derivative Test or the Second Derivative Test.

A. First Derivative Test: Check a convenient value that’s less than c to see whether $f’$ is positive or negative there; then check a convenient value that’s a greater than c to see if $f’$ is positive or negative there. As the figure below illustrates:

if the derivative switches from positive to negative, you’ve found a local maximum.

if it switches from negative to positive, you’ve found a local minimum.

if it doesn’t switch, it is neither a maximum nor a minimum; it is simply a point with a horizontal tangent (if $f'(c) = 0$), or vertical tangent (if $f'(c)$ is undefined).

B. Second Derivative Test: Compute the function’s second derivative, $f^″(x).$ Then determine the sign of the second derivative at c, $f^{″}(c)$. As the figure below illustrates:

if $f^{″}(c)$ is negative (so the function is concave down there), then $c$ is a local maximum.

if it’s positive (so the function is concave up there), then $c$ is a local minimum.

If requested, compute the value(s) of $f(c)$ to find the $y$-values of the maxima and/or minima.

If the function has endpoints ($a \le x \le b$), compute the values of $f(a)$ and $f(b)$ to see how they compare to the values of $f(c)$ you found step 4. The global (or absolute) maximum or minimum may lie at one of these endpoints.

Question 1 steps you through the process to find the relative maximum and minimum of a function. Question 2 examines the same function, but now looks for the global maximum and minimum on an interval.

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