Question

Maximum slope of the curve y=x3+3x2+9x−27 is

Answer 1

SOLUTION

CORRECT QUESTION

Maximum slope of the curve

y = - x³ + 3x² + 9x - 27

EVALUATION

Here the given equation of the Curve is

y = - x³ + 3x² + 9x - 27

Differentiating both sides with respect to x we get

y' = - 3x² + 6x + 9

So slope of the Curve = S = - 3x² + 6x + 9

Now we have to find the maximum value of S

S = - 3x² + 6x + 9

Differentiating both sides with respect to x two times we get

S' = - 6x + 6

S'' = - 6

For extremum value of S we have

S' = 0

⇒ - 6x + 6 = 0

⇒ - 6x = - 6

⇒ x = 1

Now at x = 1 we have S'' = - 6 < 0

Thus at x = 1 , S is maximum

Hence the required maximum value

$\sf{ = - {( 1)}^{3} + 3 \times {1}^{2} + 9 \times 1 - 27}$

$\sf{ = - 1 + 3 + 9 - 27}$

$\sf{ = - 16}$