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Home » Find the intervals in which the following functions are increasing or decreasing. (i) (ii)
Find the intervals in which the following functions are increasing or decreasing.
(i) f(x)=5 x^{3}-15 x^{2}-120 x+3
(ii) f(x)=x^{3}-6 x^{2}-36 x+2
CBSE | Class 12 | Exercise 17.2 | Increasing and Decreasing Functions | Maths | RD Sharma
Find the intervals in which the following functions are increasing or decreasing.
(i) f(x)=5 x^{3}-15 x^{2}-120 x+3
(ii) f(x)=x^{3}-6 x^{2}-36 x+2

Solution:

(i) Given that f(x)=5 x^{3}-15 x^{2}-120 x+3
Now with respect to x differentiating above equation, we obtain
\begin{array}{l} f^{\prime}(x)=\frac{d}{d x}\left(5 x^{3}-15 x^{2}-120 x+3\right) \\ \Rightarrow f^{\prime}(x)=15 x^{2}-30 x-120 \end{array}
For f(x) we have to find critical point, we must have
\begin{array}{l} \Rightarrow f^{\prime}(x)=0 \\ \Rightarrow 15 x^{2}-30 x-120=0 \\ \Rightarrow 15\left(x^{2}-2 x-8\right)=0 \\ \Rightarrow 15\left(x^{2}-4 x+2 x-8\right)=0 \\ \Rightarrow x^{2}-4 x+2 x-8=0 \\ \Rightarrow(x-4)(x+2)=0 \\ \Rightarrow x=4,-2 \end{array}
So, f^{\prime}(x)>0 if x<-2 and x>4 and f^{\prime}(x)<0 if -2<x<4
Therefore, f(x) increases on (-\infty,-2) \cup(4, \infty) and f(x) is decreasing on interval x \in(-2,4)

(ii) Given that f(x)=x^{3}-6 x^{2}-36 x+2
\begin{array}{l} \Rightarrow \\ f(x)=\frac{d}{d x}\left(x^{3}-6 x^{2}-36 x+2\right) \\ \Rightarrow f^{\prime}(x)=3 x^{2}-12 x-36 \end{array}
For f(x) we have to find critical point, we must have
\begin{array}{l} \Rightarrow f^{\prime}(x)=0 \\ \Rightarrow 3 x^{2}-12 x-36=0 \\ \Rightarrow 3\left(x^{2}-4 x-12\right)=0 \\ \Rightarrow 3\left(x^{2}-6 x+2 x-12\right)=0 \end{array}
\begin{array}{l} \Rightarrow x^{2}-6 x+2 x-12=0 \\ \Rightarrow(x-6)(x+2)=0 \\ \Rightarrow x=6,-2 \end{array}
So, f^{\prime}(x)>0 if x<-2 and x>6 and f^{\prime}(x)<0 if -2<x<6
Therefore, f(x) increases on (-\infty,-2) \cup(6, \infty) and f(x) is decreasing on interval x \in(-2,6)

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