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Maxima & Minima

PROBLEM SOLVING STRATEGY: Maxima & Minima
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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.

  1. Compute the function’s derivative, $f'(x)$.
  2. 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.)

  3. 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).
When the derivative switches from positive to negative at a critical point, you have a maximum. When it switches from negative to positive, you have a minimum.

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 the second derivative at a cricial point is negative, you have a maximum. (Think of an upside-down bowl.)  If it's positive, you have a minimum.  (Think of a normal bowl.)

  1. If requested, compute the value(s) of $f(c)$ to find the $y$-values of the maxima and/or minima.
  2. 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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