Chapter 9.4, Problem 33E

### Mathematical Applications for the ...

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

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=x4âˆ’4x3+9.

Formula Used:

The condition for horizontal tangent is

fâ€²(x)=0

According to sum rule of derivatives,

If

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

Then,

fâ€²(x)=uâ€²(x)+vâ€²(x)

According to power rule,

If f(x)=xn, then fâ€²(x)=nxnâˆ’1.

According to constant function rule,

If f(x)=c, then fâ€²(x)=0.

Coefficient rule for a constant c is such that, if f(x)=câ‹…u(x), where u(x) is a differentiable function of x, then fâ€²(x)=câ‹…uâ€²(x).

Calculation:

The provided function is y=x4âˆ’4x3+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,

yâ€²=d(x4)dxâˆ’d(4x3)dx+d(9)dx

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

yâ€²=d(x4)dxâˆ’d(4x3)dx+d(9)dxyâ€²=(4x4âˆ’1)âˆ’4(3x3âˆ’1)+0yâ€²=4x3âˆ’12x2

The condition for horizontal tangent is fâ€²(x)=0

Thus, for the given curve,

4x3âˆ’12x2=04x2(xâˆ’3)=0

The above equation gives two values of x for fâ€²(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