Answer :

Given equation of curve is y = –x^{3} + 3x^{2} + 9x – 27

Now applying first derivative, we get

Now applying the sum rule of differentiation and the differentiation of the constant term is 0 we get

Now applying the power rule of differentiation we get

This is the slope of the given curve.

Now we will differentiate equation (i) once again to find out the second derivative of the given curve,

Now applying the sum rule of differentiation and the differentiation of the constant term is 0 we get

Now applying the power rule of differentiation we get

Now we will find the critical point by equating the second derivative to 0, we get

-6(x-1) =0

⇒ x-1=0

⇒ x=1

Now we will differentiate equation (ii) once again to find out the third derivative of the given curve,

Now applying the sum rule of differentiation and the differentiation of the constant term is 0 we get

Now applying the power rule of differentiation we get

So the maximum slope of the given curve is at x=1

Now we will substitute x=1 in equation (i), we get

Hence the maximum slope of the curve y = –x^{3} + 3x^{2} + 9x – 27 is 12.

So the correct option is option B.

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