   Chapter 9.4, Problem 33E Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Solutions

Chapter
Section Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

In Problems 31-34, find the coordinates of points where the graph of f ( x ) has horizontal tangents. As a check, graph f ( x ) and see whether the points you found look as though they have horizontal tangents. f ( x ) = x 4 − 4 x 3 + 9

To determine

To calculate: The coordinates of point where the curve y=x44x3+9 has horizontal tangents. Also check the solution by plotting the graph of the curve.

Explanation

Given Information:

The curve is given by the function y=x44x3+9.

Formula Used:

The condition for horizontal tangent is

f(x)=0

According to sum rule of derivatives,

If

f(x)=u(x)+v(x)

Then,

f(x)=u(x)+v(x)

According to power rule,

If f(x)=xn, then f(x)=nxn1.

According to constant function rule,

If f(x)=c, then f(x)=0.

Coefficient rule for a constant c is such that, if f(x)=cu(x), where u(x) is a differentiable function of x, then f(x)=cu(x).

Calculation:

The provided function is y=x44x3+9.

The slope of the tangent at any point of the curve is given by derivative of the function.

Therefore, to calculate slope of the tangent calculate the derivative of the function.

Apply the sum rule of derivatives,

y=d(x4)dxd(4x3)dx+d(9)dx

Use the power rule, coefficient rule and constant function rule to find the derivative of the function,

y=d(x4)dxd(4x3)dx+d(9)dxy=(4x41)4(3x31)+0y=4x312x2

The condition for horizontal tangent is f(x)=0

Thus, for the given curve,

4x312x2=04x2(x3)=0

The above equation gives two values of x for f(x)=0

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Evaluate the integral. 30. dxex1e2x

Calculus: Early Transcendentals

Explain why the formula for sample variance is different from the formula for population variance.

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

True or False: converges mean exists.

Study Guide for Stewart's Multivariable Calculus, 8th 