Explanation Given: The function is f ( x ) = − x 3 + 3 x 2 + 9 x − 10 . Definition used: A function f ( x ) said to be symmetric about the y -axis if f ( − x ) = f ( x ) Calculation: The given function is f ( x ) = − x 3 + 3 x 2 + 9 x − 10 . Domain: The given function f ( x ) = − x 3 + 3 x 2 + 9 x − 10 is defined for all real values. Thus, the domain of the function f ( x ) = − x 3 + 3 x 2 + 9 x − 10 is ( − ∞ , ∞ ) . Symmetry: The given function is f ( x ) = − x 3 + 3 x 2 + 9 x − 10 . Substitute x = − x in f ( x ) , f ( − x ) = − ( − x ) 3 + 3 × ( − x ) 2 + 9 ( − x ) − 10 = x 3 + 3 x 2 − 9 x − 10 Thus, f ( − x ) ≠ f ( x ) . Therefore, from the definition stated above and conclude that the polynomial has no symmetry. Intercept: The function f ( x ) intercepts y -axis at x = 0 . Substitute the value x = 0 in f ( x ) , f ( 0 ) = − ( 0 ) 3 + 3 ( 0 ) 2 + 9 ( 0 ) − 10 = − 10 That is the function f ( x ) intercepts y -axis at the point ( 0 , − 10 ) . Asymptote: The function f ( x ) = − x 3 + 3 x 2 + 9 x − 10 has no asymptote. Critical point: The given function is f ( x ) = − x 3 + 3 x 2 + 9 x − 10 . Differentiate f ( x ) with respect to x , f ′ ( x ) = − 3 x 2 + 6 x + 9 To obtain critical points, f ′ ( x ) = 0 , − 3 x 2 + 6 x + 9 = 0 − 3 ( x 2 − 2 x − 3 ) = 0 ( x + 1 ) ( x − 3 ) = 0 Thus, x = 3 and x = − 1 . Thus, the critical points are at x = 3 and x = − 1 . Substitute x = 3 and x = − 1 in f ( x ) = − x 3 + 3 x 2 + 9 x − 10 and obtain the critical points as ( − 1 , − 15 ) and ( 3 , 17 ) . Relative Maxima and Relative Minima : The critical points are at x = 3 and x = − 1 . Substitute x = − 2 in f ′ ( x ) = − 3 x 2 + 6 x + 9 , f ′ ( − 2 ) = − 3 × ( − 2 ) 2 + 6 × ( − 2 ) + 9 = − 12 − 12 + 9 = − 15 < 0 Thus, the function f ( x ) is decreasing on the interval ( − ∞ , − 1 )

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

Concept explainers

Transformation of Graphs

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Exponential Functions

The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, c…

Videos

Question

Chapter 13.4, Problem 1YT

To determine

To sketch: The graph of the function.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.3, Problem 96E

Chapter 13.4, Problem 2YT

Students have asked these similar questions

Find the ane sided limit lim 2 x+1-3x-3

4 44 3+ 2+ -4-3 -2 -1 2 3 4 -2 -3. Find the values at X which the graph is discontinuous and categorize them.

Find the following limit lim x-4 X-72 3x-6

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 graph of the function.

Solution Summary: The function f(x) is symmetric about the y -axis if it is defined for all real values.

Concept explainers

Videos

To sketch: The graph of the function.

Want to see the full answer?

Chapter 13 Solutions