UPSC NDA I 2021 Mathematics Solved Paper

Concept:
Slope of a curve y=f(x) at a point is the value of its first derivative at that point.
i.e. m=f(x)
Calculation:
The slope of the curve y=−x3+3x2+2x−27 will be given by:
m=y′=‌

d

dx

(−x3+3x2+2x−27)=−3x2+6x+2
In order to maximize the slope, we must have m′=0 and m′′<0. Now, m′=0.
⇒‌

d

dx

(−3x2+6x+2)=−6x+6=0
⇒x=1
And m′′=−6<0.
Therefore, the slope is maximum at x=1.