Question

The maximum slope of curve \[y = - {x^3} + 3{x^2} + 9x - 27\] is
A) 0
B) 12
C) 32
D) 16

Answer 1

Hint:
To find the maximum slope of the curve, first, we’ll differentiate the given curve. then we’ll find what point-slope will be maximum and in the end, we’ll put the value to get the maximum slope.

Complete step by step solution:
Given the curve is \[y = - {x^3} + 3{x^2} + 9x - 27\].
To get the equation for the slope we will differentiate it once.
\[ \Rightarrow y' = - 3{x^2} + 6x + 9\]
The equation above will give us the value of the slope of the curve. But we are asked to find the maximum slope.
For that purpose, we need the maxima or minima point of the curve.
To differentiate it one more time and equate it to 0.
\[ \Rightarrow y'' = - 6x + 6\]
Now equate it to 0.
\[ - 6x + 6 = 0\]
\[ \Rightarrow x = 1\]
Now to find whether the curve obtains maxima or minima at x=1, again differentiate the curve.
\[ \Rightarrow y''' = - 6\]
Thus
 \[
   \Rightarrow y'''\left( 1 \right) = - 6 \\
   \Rightarrow y'''\left( 1 \right) < 0 \\
 \]
This is the point of maxima. Now the maximum slope of the curve is
\[
   \Rightarrow y'\left( 1 \right) = - 3{\left( 1 \right)^2} + 6 \times 1 + 9 \\
   \Rightarrow y'\left( 1 \right) = - 3 + 15 \\
   \Rightarrow y'\left( 1 \right) = 12 \\
\]
Thus the maximum slope of the curve is 12.

Thus, option B is correct.

Note:
Once differentiated function (equation of curve here) will give you the equation for slope only. Double differentiate will help you in finding the point of maxima or minima. So use differentiation according to the requirement.