Chapter 3: Q. 42 (page 275)

Use a sign chart for $f''$ to determine the intervals on which each function f in Exercises 41–52 is concave up or concave down, and identify the locations of any inflection points. Then verify your algebraic answers with graphs from a calculator or graphing utility
$f (x) = (x - 3)^{3} (x - 1)$

Short Answer

Expert verified

The function is concave up on both $(- \infty, 2), (3, \infty)$ and concave down on $(2, 3) .$

Step by step solution

Step 1. Given Information.

The given function is $f (x) = (x - 3)^{3} (x - 1)$

Step 2. Second derivative.

On differentiating, we get,

$f^{'} (x) = \frac{d}{d x} (x - 3)^{3} (x - 1) = (\frac{d}{d x} (x - 3)^{3}) (x - 1) + (x - 3)^{3} (\frac{d}{d x} (x - 1)) = 3 (x - 3)^{2} (x - 1) + (x - 3)^{3} = 2 (x - 3)^{2} (2 x - 3) f^{''} (x) = \frac{d}{d x} 2 (x - 3)^{2} (2 x - 3) = 2 (\frac{d}{d x} (x - 3)^{2} (2 x - 3)) = 2 (2 (x - 3) (2 x - 3) + 2 (x - 3)^{2}) = 12 (x - 3) (x - 2)$

Step 3. Sign chart.

Now,

$f'' (x) = 0 w h e n x = 2 and x = 3,$ .

Therefore, the sign chart will be,

The function has inflexion points $x = 2, 3$ .

$Thefunctionis concave up on both (- \infty, 2), (3, \infty) and concave down on (2, 3)$

Step 4. Verification.

The graph of the function is,

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Use a sign chart for $f''$ to determine the intervals on which each function f in Exercises 41–52 is concave up or concave down, and identify the locations of any inflection points. Then verify your algebraic answers with graphs from a calculator or graphing utility
$f (x) = (x - 3)^{3} (x - 1)$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Second derivative.

Step 3. Sign chart.

Step 4. Verification.

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Use a sign chart for f''to determine the intervals on which each function f in Exercises 41–52 is concave up or concave down, and identify the locations of any inflection points. Then verify your algebraic answers with graphs from a calculator or graphing utility f(x)=(x-3)3(x-1)

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Second derivative.

Step 3. Sign chart.

Step 4. Verification.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Calculus

Theoretical and Mathematical Physics

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Use a sign chart for $f''$ to determine the intervals on which each function f in Exercises 41–52 is concave up or concave down, and identify the locations of any inflection points. Then verify your algebraic answers with graphs from a calculator or graphing utility
$f (x) = (x - 3)^{3} (x - 1)$