Chapter 3: Q. 40 (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.
$f (x) = x^{2} (x - 1) (x + 1)$

Short Answer

Expert verified

Ans: The local extrema value of f(x) are

Max at $x = 0$

Min at $x = \pm \sqrt{\frac{1}{2}}$

Step by step solution

Step 1. Given Information

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

Step 2. Finding the derivative of the function

Rewriting the function by simplifying it,

$f (x) = x^{2} (x - 1) (x + 1) = x^{2} (x^{2} - 1) = x^{4} - x^{2} f' (x) = 4 x^{3} - 2 x l e t, f' (x) = 0 4 x^{3} - 2 x = 0 2 x (2 x^{2} - 1) = 0 2 x = 0 x = 0 2 x^{2} - 1 = 0 2 x^{2} = 1 x = \pm \sqrt{\frac{1}{2}} x = 0, \pm \sqrt{\frac{1}{2}}$

Step 3. Substituting the values into the function equation:

$x = - 2, \frac{1}{2}, 2 i n t o t h e f' (x) = x^{4} - x^{2} f' (- 2) = {(- 2)}^{4} - {(- 2)}^{2} = 12 > 0 f' (2) = {(2)}^{4} - {(2)}^{2} = 12 > 0 f' (\frac{1}{2}) = {(\frac{1}{2})}^{4} - {(\frac{1}{2})}^{2} = \frac{- 3}{16} < 0$

Step 4. Finding local extrema of the function on the number line(Sign chart):

Step 5. 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.
$f (x) = x^{2} (x - 1) (x + 1)$

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. Finding local extrema of the function on the number line(Sign chart):

Step 5. Verifying algebraic answers with graphs :

One App. One Place for Learning.

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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)=x2(x-1)(x+1)

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. Finding local extrema of the function on the number line(Sign chart):

Step 5. Verifying algebraic answers with graphs :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Decision Maths

Probability and Statistics

Geometry

Discrete Mathematics

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