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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

For what values of the constants a and b is (1, 3) a point of inflection of the curve y = a x 3 + b x 2 ?

To determine

To find:

The values of the constants a and b so that (1, 3) is a point of inflection of the curve

 y=ax3+bx2

Explanation

1) Concept:

Use the concept of inflection point.

 If f''x>0, the function is concave up and f''x<0, the function is concave down in that particular interval and f''x=0 gives the values of inflection points.

2) Given:

y=ax3+bx2 and point of inflection (1, 3)

3) Calculation:

Here, (1, 3) is a point of inflection, so it should lie on a curve.

Thus, it satisfies the equation of the curve y=ax3+bx2.

So substitute x=1 and y= 3 in the given equation, it becomes

a+b=3……………. (1)

Now find the second derivative with respect to x.

Differentiate y twice with respect to x

y=ax3+bx2

y'=3ax2+2bx

Again differentiate y' with respect to x, it becomes

y"=6ax+2b

Substitute x=1 and then set y" = 0

0=6a+2b

6a+2b=0

Factor 2 from left side of the equation,

3a+b=0…………

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