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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Find the highest and lowest points on the curve x2 + xy + y2 = 12.

To determine

To find: The highest and lowest points on the curve x2+xy+y2=12 .

Explanation

To find the highest and lowest points it is enough to find maxima and minima of the curve x2+xy+y2=12 .

Differentiate the equation of the curve with respect to x.

x2+xy+y2=122x+dydx+y+2ydydx=0

At the critical points value of first derivative is 0. So, set dydx=0 and find the critical point.

2x+0+y+2y0=02x+y=0y=2x

Substitute y=2x in the equation of the curve.

x2+xy+y2=12x2+x(2x)+(2x)2=123x2=12x=±2

Thus, the value of y is, y=±4 .

Therefore, the critical points are (2,4) and (2,4) .

To determine at which points the curve has maxima and minima, differentiate the equation of the curve again with respect to x.

2x+dydx+y+2ydydx=02+xd2ydx2+dydx+dydx+2yd2ydx2+2(dydx)2=0

Substitute dydx=0 ,

2+xd2y

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