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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+36 x+3 x^{2}-2 x^{3}
(ii) f(x)=8+36 x+3 x^{2}-2 x^{3}
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+36 x+3 x^{2}-2 x^{3}
(ii) f(x)=8+36 x+3 x^{2}-2 x^{3}

Solution:

(i) Given that f(x)=5+36 x+3 x^{2}-2 x^{3}
f(x)=\frac{d}{d x}\left(5+36 x+3 x^{2}-2 x^{3}\right)
\Rightarrow f^{\prime}(x)=36+6 x-6 x^{2}
For f(x) now we have to find critical point, we must have
\begin{array}{l} \Rightarrow f^{\prime}(x)=0 \\ \Rightarrow 36+6 x-6 x^{2}=0 \\ \Rightarrow 6\left(-x^{2}+x+6\right)=0 \\ \Rightarrow 6\left(-x^{2}+3 x-2 x+6\right)=0 \\ \Rightarrow-x^{2}+3 x-2 x+6=0 \\ \Rightarrow x^{2}-3 x+2 x-6=0 \\ \Rightarrow(x-3)(x+2)=0 \\ \Rightarrow x=3,-2 \end{array}
So, f^{\prime}(x)>0 if -2<x<3 and f^{\prime}(x)<0 if x<-2 and x>3
Therefore, f(x) increases on x \in(-2,3) and f(x) is decreasing on interval (-\infty,-2) \cup(3, \infty),

(ii) Given that f(x)=8+36 x+3 x^{2}-2 x^{3}
On differentiating with respect to x we get,
\begin{array}{l} f(x)=\frac{d}{d x}\left(8+36 x+3 x^{2}-2 x^{3}\right) \\ \Rightarrow f^{\prime}(x)=36+6 x-6 x^{2} \end{array}
For f(x) we have to find critical point, we must have
\begin{array}{l} \Rightarrow f^{\prime}(x)=0 \\ \Rightarrow 36+6 x-6 x^{2}=0 \\ \Rightarrow 6\left(-x^{2}+x+6\right)=0 \\ \Rightarrow 6\left(-x^{2}+3 x-2 x+6\right)=0 \\ \Rightarrow-x^{2}+3 x-2 x+6=0 \\ \Rightarrow x^{2}-3 x+2 x-6=0 \\ \Rightarrow(x-3)(x+2)=0 \\ \Rightarrow x=3,-2 \end{array}
So, f^{\prime}(x)>0 if -2<x<3 and f^{\prime}(x)<0 if x<-2 and x>3
Therefore, f(x) increases on x \in(-2,3) and f(x) is decreasing on interval (-\infty, 2) \cup(3, \infty)

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