   Chapter 10, Problem 25RE ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# In Problems 25 and 26, a function and its graph are given.(a) Use the graph to determine (estimate) x-values where f ' ( x ) > 0, where f ’ ( x ) < 0, and where f ' ( x ) = 0.(b) Use the graph to determine x-values where f " ( x ) > 0, where f " ( x ) < 0, and where f " ( x ) = 0.(c) Check your conclusions to (a) by finding f ' ( x ) and graphing it with a graphing calculator.(d) Check your conclusions to (b) by finding f"(x) and graphing it with a graphing calculator. f ( x ) = x 3 − 4 x 2 + 4 x (a)

To determine

The values of x for which f(x)>0,f(x)<0 and f(x)=0 from the graph of the function f(x)=x34x2+4x.

Explanation

Given Information:

The provided function is f(x)=x34x2+4x, and the graph of the function is,

Explanation:

Consider the provided function f(x)=x34x2+4x.

Use the simple power rule to differentiate the provided function,

ddx[f(x)]=ddx(x34x2+4x)f(x)=3x28x+4

The values of f(x) is zero when there is a turning point in the graph of f(x)=x34x2+4x.

The value of f(x) is positive when the function f(x)=x34x2+4x increases as the value of x increases. That means the graph of the provided function has increasing nature.

Similarly, the value of f(x) is negative when the function f(x)=x34x2+4x decreases as the value of x increases

(b)

To determine

The values of x for which f(x)>0,f(x)<0 and f(x)=0 from the graph of the function f(x)=x34x2+4x.

(c)

To determine

Whether the conclusions made in part (a) are correct by graphing the derivative of the function y=x34x2+4x by graphing calculator.

(d)

To determine

Whether the conclusions made in part (b) are correct by graphing the second derivative of the function y=x34x2+4x with a graphing calculator.

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