Mathematics 150AL, Fall 2003

Assignment 5 -- Local and Global Extrema

An extremum of a function is either its maximum or its minimum. A function $f$ has a local maximum at a point $x_{0}$ if $f(x_{0})\geq f(x)$ for all values of $x$ close to $x_{0}$. I.e. the value of $f$ at $x_{0}$ is larger than or equal to the values of $f$ at neighboring points. At a local minimum the value of the function is smaller than at neighboring points.

One way to find and classify the $x-$values where the extrema of a function occur is to construct first and second derivative sign charts for the function. The first derivative sign chart illustrates the intervals for which the first derivative is positive, negative, zero, and undefined. Similarly, the second derivative sign chart illustrates the intervals for which the second derivative is positive, negative, zero, and undefined.

Example

Let $f(x)=x^{2}+x-1$. Then the first derivative of $f$ is MATHso the derivative equals $0$ at $x=-\frac{1}{2}$. It is positive for $x>-\frac{1}{2}$ and negative for $x<-\frac{1}{2}$. Thus the first derivative sign chart is:

MATH

Similarly, the second derivative of $f$ is MATH. Thus, the second derivative never equals $0$ and is positive for all real $x$. The second derivative sign chart is:

MATH

The $x-$values for which the first derivative equals $0$ or is undefined are called the critical points of $f$. If the first derivative changes sign from positive to negative about an $x-$value for which the first derivative is $0$, the graph has a local maximum at that $x-$value. If the first derivative changes sign from negative to positive about an $x-$value for which the first derivative is $0$, the graph has a local minimum at that $x-$value. Another way to determine whether there is a local maximum or a local minimum at the critical point is to look at the sign of the second derivative. If the second derivative is negative at that $x-$value, the graph has a local maximum at that $x-$value and if the second derivative is positive at that $x-$value, the graph has a local minimum at that $x-$value.

When the second derivative changes sign about an $x-$value where the second derivative is $0$, the graph has an inflection point at that $x-$value.

In our example, the sign of the first derivative changes from negative to positive about $x=-\frac{1}{2}$, so the function has a local minimum of MATH at $x=-\frac{1}{2}$. (Equivalently, the second derivative is positive at $x=-\frac{1}{2}$ so the graph has a local minimum there.) There are no inflection points.

  1. Let MATH.

    1. Construct first and second derivative sign charts for $g$.

    2. Use the first derivative sign chart to find all critical points of $g$.

    3. Use the second derivative sign chart to classify each critical point of $g$ as a local minimum or local maximum.

    4. The smallest local minimum of a function is said to be the global minimum and the largest local maximum is said to be the global maximum. Determine whether $g$ has a global minimum or global maximum.

    5. Use the second derivative sign chart to find all points of inflection of $g$.

    6. If you haven't already, graph $g$ to confirm your results above. (Be sure to modify the viewing rectangle in order to see the shape of the graph near the critical points.)

  2. Now let $g(x)=2x^{4}-3x^{3}$ for $-1\leq x\leq 2$. Use your work in problem 1 to find all local and global minima and maxima of $g$ on this restricted domain. (Hint: Endpoints of the domain are candidates. A graph may be helpful.)

    1. What if you change the domain for $g$ to be $-1<x\leq 2$? How does your answer to part (a) change?

  3. Let MATH

    1. Construct first and second derivative sign charts for $h$.

    2. Use the first derivative sign chart to find all critical points of $h$.

    3. Use the second derivative sign chart to classify each critical point of $h$ as a local minimum or local maximum.

    4. Determine whether $h$ has a global minimum or global maximum.

    5. Use the second derivative sign chart to find all points of inflection of $h$.

    6. If you haven't already, graph $h$ to confirm your results above. (Be sure to modify the viewing rectangle in order to see the shape of the graph near the critical points.)

  4. Let MATH for $-1\leq x<1$

    1. Use any method to find all points of inflection and all local and global minima and maxima of $l$. (If you can tell just by looking at the graph, you need not use calculus to determine the answers. However, be sure to explain how the graph helped you.)

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