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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Use the methods of this section to sketch the curve y   =   x 3 3 a 2 x + 2 a 3 , where a is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?

To determine

To sketch:

The given curve by using the given method to interpret what do the members of this family of curves have in common and how they differ from each other.

Explanation

1) Concept:

i. Increasing or Decreasing test:

If f'x>0 then f is increasing

If f'x<0 then f is decreasing

ii. Concavity of function:

If f''x>0 then f is concave upward.

If f''x<0 then f is concave downward.

2) Given:

y=x3-3a2x+2a3,  a>0

3) Calculation:

Consider the given function y= x3-3a2x+2a3,  a>0.

Therefore, the y-intercept is

y0=03-3a20+2a3

y0=2a3

For finding critical numbers, differentiate y with respect to x,

y'x=3x2-3a2=3x2-a2=3 (x+a)(x-a)

The critical numbers are,

y'x=3x+ax-a=0

x=a or x= -a

Now domain of above function is (-, )

We split domain into the intervals according to critical numbers

By using Increasing or decreasing test,

On interval -,- a, y'-2a=3-2a+a-2a-a=3-a-3a=9a2>0

Therefore, y is increasing.

On interval -a, a, y'0=30+a0-a=3a-a=-3a2<0

Therefore, y is decreasing.

On interval a,, y'2a=32a+a2a-a=33aa=9a2>0

Therefore, y is increasing.

Therefore, y is increasing on -, -a(a, ) and f is decreasing on (-a, a).

As function changes from increasing to decreasing at a, therefore there is local maxima at x=-a, also at x=a function is changes from decreasing to increasing so there is local minima at x=a

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