Chapter 3: Q. 42 (page 261)

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.
$f (x) = \frac{{(x - 1)}^{2}}{x + 2}$

Short Answer

Expert verified

Ans: The local minimum of the function f(x) is at $x = 1$

The local maximum of the function f(x) is at $x = - 5$

Step by step solution

Step 1. Given Information:

$f (x) = \frac{{(x - 1)}^{2}}{x + 2}$

Step 2. Finding the derivative of the function

Rewriting the function by simplifying it,

$f (x) = \frac{{(x - 1)}^{2}}{x + 2} f' (x) = \frac{(x + 2) 2 (x - 1) - 1 - (x - 1)^{2} - 1}{(x + 2)^{2}} f' (x) = \frac{(x + 2) (2 x - 2) - (x^{2} - 2 x + 1)}{(x + 2)^{2}} f' (x) = \frac{2 x^{2} - 2 x + 4 x - 4 - x^{2} + 2 x - 1}{(x + 2)^{2}} f' (x) = \frac{x^{2} + 4 x - 5}{(x + 2)^{2}} l e t, f' (x) = 0 ∴ \frac{x^{2} + 4 x - 5}{(x + 2)^{2}} = 0 \begin{array}{r} x^{2} + 4 x - 5 = 0 \\ x^{2} + 5 x - x - 5 = 0 \\ x (x + 5) - (x + 5) = 0 \\ (x + 5) (x - 1) = 0 \end{array} s o, t h e c i t i c a l p o i n t s a r e x = - 5, 1$

Step 3. Substituting the values into the function equation:

$f (- 5) = \frac{{(- 5 - 1)}^{2}}{- 5 + 2} = \frac{{(- 6)}^{2}}{- 3} = \frac{36}{- 3} = - 12 < 0 f (1) = \frac{{(1 - 1)}^{2}}{1 + 2} = \frac{{(0)}^{2}}{3} = 0 = 0 ∴ t h e l o c a l m i n i m u m a t x = 1 : a n d l o c a l m a x i m u m a t x = - 5$

Step 4. Verifying algebraic answers with graphs :

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Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.
$f (x) = \frac{{(x - 1)}^{2}}{x + 2}$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Finding the derivative of the function

Step 3. Substituting the values into the function equation:

Step 4. Verifying algebraic answers with graphs :

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Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.f(x)=(x-1)2x+2

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Finding the derivative of the function

Step 3. Substituting the values into the function equation:

Step 4. Verifying algebraic answers with graphs :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Statistics

Pure Maths

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.
$f (x) = \frac{{(x - 1)}^{2}}{x + 2}$