Chapter 11: Q. 45 (page 890)

Find the value(s) of x where the curvature is greatest on the graph of $y = x^{3} .$

Short Answer

Expert verified

When the value of xis $\pm \frac{1}{{(45)}^{\frac{1}{4}}}$ the curvature is greatest on the graph of $y = x^{3} .$

Step by step solution

Step 1. Given Information.

The given graph is $y = x^{3} .$

Step 2. Find the value.

To find the value(s) of x where the curvature is greatest on the graph of $y = x^{3},$ we will use the formula for the curvature in the plane.

The curvature in the plane is $k = \frac{|f'' (x)|}{{(1 + {(f' (x))}^{2})}^{\frac{3}{2}}} .$

Now, $y = f (x) = x^{3} .$

So,

$f (x) = x^{3} f' (x) = 3 x^{2} f'' (x) = 6 x Now, the curvature of the graph of f is k = \frac{|f'' (x)|}{{(1 + {(f' (x))}^{2})}^{\frac{3}{2}}} k = \frac{|6 x|}{{(1 + {(3 x^{2})}^{2})}^{\frac{3}{2}}} k = \frac{6 x}{{(1 + 9 x^{4})}^{\frac{3}{2}}}$

Step 3. Use the first derivative test.

Now, to find the value of x on the graph of $y = x^{3}$ where the curvature is greatest, we will apply the derivative test.

So,

$Let g (x) = \frac{6 x}{{(1 + 9 x^{4})}^{\frac{3}{2}}} g' (x) = \frac{{(1 + 9 x^{4})}^{\frac{3}{2}} (6) - (6 x) \frac{3}{2} {(1 + 9 x^{4})}^{\frac{1}{2}} (36 x^{3})}{{(1 + 9 x^{4})}^{3}} g' (x) = \frac{6 {(1 + 9 x^{4})}^{\frac{3}{2}} - 324 x^{4} \sqrt{(1 + 9 x^{4})}}{{(1 + 9 x^{4})}^{3}} Put g' (x) = 0 0 = \frac{6 {(1 + 9 x^{4})}^{\frac{3}{2}} - 324 x^{4} \sqrt{(1 + 9 x^{4})}}{{(1 + 9 x^{4})}^{3}} 0 = 6 {(1 + 9 x^{4})}^{\frac{3}{2}} - 324 x^{4} \sqrt{(1 + 9 x^{4})} 0 = 6 \sqrt{1 + 9 x^{4}} [(1 + 9 x^{4}) - 54 x^{4}] 0 = 1 + 9 x^{4} - 54 x^{4} 0 = 1 - 45 x^{4} \frac{1}{45} = x^{4} x = \pm \frac{1}{{(45)}^{\frac{1}{4}}}$

Thus, when $x = \pm \frac{1}{{(45)}^{\frac{1}{4}}}$ the curvature is greatest on the graph of $y = x^{3} .$

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Find the value(s) of x where the curvature is greatest on the graph of $y = x^{3} .$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the value.

Step 3. Use the first derivative test.

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Find the value(s) of x where the curvature is greatest on the graph of y=x3.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the value.

Step 3. Use the first derivative test.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

Logic and Functions

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Find the value(s) of x where the curvature is greatest on the graph of $y = x^{3} .$