UPSC NDA Math Model Paper 2

We know that, . If f′(x) > 0 at each point in an interval I, then the function is said to be increasing on I. f′(x) < 0 at each point in an interval I, then the function is said to be decreasing on I
Given ,f(x)=x3−6x2+9x+10
Differentiating, we get
f′(x)=3x2−12x+9
f(x) is increasing function
⇒f′(x)≥0
⇒3x2−12x+9≥0
⇒x2−4x+3≥0.
⇒(x−3)(x−1)≥0
Hence, x∈(−∞,1]∪[3,∞)
The interval in which the function f(x)=x3−6x2+9x+10 is increasing in (−∞,1]∪[3,∞)