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Question

Find the minimum and maximum value of the function y=x33x2+6. Find the values of y at which it occurs.


A

Maxima at x = 0, Maximum value = 6

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B

Maxima at x = 2, Maximum value = 2

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C

Minima at x=0,minimum value = 6

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D

Minima at x=2,minimum value = 6

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Solution

The correct options are
A

Maxima at x = 0, Maximum value = 6


D

Minima at x=2,minimum value = 6


Given y=x33x2+6
Differentiating y w.r.t. 'x', dydx=3x26x
Putting dydx=0, we will get the values at which function is maximum or minimum
3x26x=0x(3x6)=0
x=0,+2
To distinguish values of x as the point of maximum or minimum, we need 2nd derivative of the function.
d2ydx2=6x6; Now(d2ydx2)x=0=6<0
At x=0It is maximum(d2ydx2)x=2=6(2)6=6>0At x=2It is minimum
Hence x=0 is a point of maximum and x = 2 is a point of minimumSo, maximum value of y = 033.0+6=6minimum value of y=(2)33(2)2+6=2


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