376
Chapter 4 | Applications of Derivatives
4.3 EXERCISES 90. In precalculus, you learned a formula for the position of the maximum or minimum of a quadratic equation y = ax 2 + bx + c , which was h = − b (2 a ) . Prove this formula using calculus. 91. If you are finding an absolute minimum over an interval [ a , b ], why do you need to check the endpoints? Draw a graph that supports your hypothesis. 92. If you are examining a function over an interval ( a , b ), for a and b finite, is it possible not to have an absolute maximum or absolute minimum? 93. When you are checking for critical points, explain why you also need to determine points where f ' ( x ) is undefined. Draw a graph to support your explanation. 94. Can you have a finite absolute maximum for y = ax 2 + bx + c over (−∞, ∞)? Explain why or why not using graphical arguments. 95. Can you have a finite absolute maximum for y = ax 3 + bx 2 + cx + d over (−∞, ∞) assuming a is non-zero? Explain why or why not using graphical arguments. 96. Let m be the number of local minima and M be the number of local maxima. Can you create a function where M > m +2? Draw a graph to support your explanation. 97. Is it possible to have more than one absolute maximum? Use a graphical argument to prove your hypothesis. 98. Is it possible to have no absolute minimum or maximum for a function? If so, construct such a function. If not, explain why this is not possible. 99. [T] Graph the function y = e ax . For which values of a , on any infinite domain, will you have an absolute minimum and absolute maximum? For the following exercises, determine where the local and absolute maxima and minima occur on the graph given. Assume the graph represents the entirety of each function.
100.
101.
102.
103.
For the following problems, draw graphs of f ( x ), which is continuous, over the interval [−4, 4] with the following properties:
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online