Explanation Definition used: First derivative test: Suppose c to be a critical number for a function f . Assume that f is continuous on ( a , b ) and differentiable on ( a , b ) except at c , and c is the only critical number of f in ( a , b ) . f ( c ) is a relative maximum of f at c if f ′ ( x ) > 0 in the interval ( a , c ) and f ′ ( x ) < 0 in the interval ( c , b ) . f ( c ) is a relative minimum f at c if f ′ ( x ) < 0 in the interval ( a , c ) and f ′ ( x ) > 0 in the interval ( c , b ) . Relative maximum: A function is said to have a relative maximum f ( c ) at c if f ′ ( x ) > 0 in the interval ( a , c ) and f ′ ( x ) < 0 in the interval ( c , b ) . Relative minimum: A function is said to have a relative maximum f ( c ) at c if f ′ ( x ) < 0 in the interval ( a , c ) and f ′ ( x ) > 0 in the interval ( c , b ) . Calculation: Use online graphing calculator to plot the graph of the function f ( x ) = − x 5 − x 4 + 2 x 3 − 25 x 2 + 9 x + 12 as shown below in Figure 1. From Figure 1, it is observed that, the relative extrema are at the values x = 0.183 and x = − 2.703 . The relative maximum of 12.821 occurs at x = 0.183 and the relative minimum of − 143.572 occurs at x = − 2.703 . Now, to verify this, obtain the derivative of the function f ( x ) = − x 5 − x 4 + 2 x 3 − 25 x 2 + 9 x + 12 as follows

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 13.2, Problem 38E

To determine

To sketch: The function f(x)=−x5−x4+2x3−25x2+9x+12 in the window [−10,10] by[−15,30] to find all the relative extrema and to confirm the derivative with the help of calculator to solve f′(x)=0 .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.2, Problem 37E

Chapter 13.2, Problem 39E

Students have asked these similar questions

Explain the focus and reasons for establishment of 12.5.1(lim(n->infinite) and sigma of k=0 to n)

Explain the focus and reasons for establishment of 12.5.3 about alternating series. and explain the reason why (sigma k=1 to infinite)(-1)k+1/k = 1/1 - 1/2 + 1/3 - 1/4 + .... converges.

Explain the key points and reasons for the establishment of 12.3.2(integral Test)

Chapter 13 Solutions

Finite Mathematics and Calculus with Applications (10th Edition)

Ch. 13.1 - Find where the function is increasing and...Ch. 13.1 - Prob. 2YT Ch. 13.1 - Prob. 3YT Ch. 13.1 - Prob. 4YT Ch. 13.1 - Prob. 1WE Ch. 13.1 - Prob. 2WE Ch. 13.1 - Prob. 3WE Ch. 13.1 - Prob. 4WE Ch. 13.1 - Prob. 5WE Ch. 13.1 - Prob. 6WE

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The function f ( x ) = − x 5 − x 4 + 2 x 3 − 25 x 2 + 9 x + 12 in the window [ − 10 , 10 ] by [ − 15 , 30 ] to find all the relative extrema and to confirm the derivative with the help of calculator to solve f ′ ( x ) = 0 .

Solution Summary: The author illustrates the function f(x) as a critical number and the relative extrema of 12.82 and 143.572.

Videos

To sketch: The function f(x)=−x5−x4+2x3−25x2+9x+12 in the window [−10,10] by[−15,30] to find all the relative extrema and to confirm the derivative with the help of calculator to solve f′(x)=0 .

Want to see the full answer?

Chapter 13 Solutions