Chapter 4.6, Problem 104E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Graphical Reasoning Consider the function f ( x ) = 1 2 ( a x ) 2 − a x ,     a ≠ 0 .(a) Determine the changes (if any) in the intercepts, extrema, and concavity of the graph of f when a is varied.(b) In the same viewing window, use a graphing utility to graph the function for four different values of a.

(a)

To determine
The effect on the intercepts, concavity and extremas of the graph of the provided function f(x)=12(ax)2ax with a0 when the value of a changes.

Explanation

Consider the function:

f(x)=12(ax)2âˆ’ax

The x and y intercepts can be computed by equating f(x) with 0.

12(ax)2âˆ’ax=0x=0,2a

Thus, the x-intercepts are (0,0) and (2a,0).

And,

f(0)=12(a(0))2âˆ’a(0)=0

Thus, the y-intercept is (0,0). It is clear that one x-intercept would vary with a.

Differentiate the function and equate the derivative to zero to obtain the critical point.

f'(x)=a(axâˆ’1)a(axâˆ’1)=0axâˆ’1=0x=1a

Now check the sign of the second derivative at 1a

(b)

To determine

To graph: The function f(x)=12(ax)2ax for 4 values of a.

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