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Question

Find the maximum slope of the curve y=-x3+3x2+2x-27.

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Solution

Given:y=-x3+3x2+2x-27 ...1Slope= dydx=-3x2+6x+2Now,M=-3x2+6x+2dMdx=-6x+6For maximum or minimum values of M, we must havedMdx=0-6x+6=06x=6x=1Substituing the value of x in eq. 1, we gety=-13+3×12+2×1-27=-23d2Mdx2=-6<0So, the slope is maximum when x=1 and y=-23. At 1, -23: Maximum slope=-312+61+2=-3+6+2=5

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