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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)=6-9 x-x^{2}
(ii) f(x) = 2x^3 - 12x^2 + 18x + 15
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)=6-9 x-x^{2}
(ii) f(x) = 2x^3 - 12x^2 + 18x + 15

Solution:

(i) Given that f(x)=6-9 x-x^{2}
\begin{array}{l} \Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(6-9 x-x^{2}\right) \\ \Rightarrow f^{\prime}(x)=-9-2 x \end{array}
For f(x) to be increasing, we must have
\begin{array}{l} \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})>0 \\ \Rightarrow-9-2 \mathrm{x}>0 \\ \Rightarrow-2 \mathrm{x}>9 \\ \Rightarrow \mathrm{x}<-\frac{9}{2} \\ \Rightarrow \mathrm{x}<-\frac{9}{2} \\ \Rightarrow \mathrm{x} \in\left(-\infty,-\frac{9}{2}\right) \end{array}
Therefore f(x) is increasing on interval \left(-\infty,-\frac{9}{2}\right)
Again, for f(x) to be decreasing, we must have
\begin{array}{l} f^{\prime}(x)<0 \\ \Rightarrow-9-2 x<0 \\ \Rightarrow-2 x<9 \\ \Rightarrow x>-\frac{9}{2} \\ \Rightarrow x>-\frac{9}{2} \\ \Rightarrow x \in\left(-\frac{9}{2}, \infty\right) \end{array}
Therefore f(x) is decreasing on interval \mathrm{x} \in\left(-\frac{9}{2}, \infty\right)

(ii) Given that f(x)=2 x^{3}-12 x^{2}+18 x+15
\begin{array}{l} \Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(2 x^{3}-12 x^{2}+18 x+15\right) \\ \Rightarrow f^{\prime}(x)=6 x^{2}-24 x+18 \end{array}
For \mathrm{f}(\mathrm{x}) we have to find critical point, we must have
\begin{array}{l} \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=0 \\ \Rightarrow 6 \mathrm{x}^{2}-24 \mathrm{x}+18=0 \\ \Rightarrow 6\left(\mathrm{x}^{2}-4 \mathrm{x}+3\right)=0 \\ \Rightarrow 6\left(\mathrm{x}^{2}-3 \mathrm{x}-\mathrm{x}+3\right)=0 \\ \Rightarrow 6(\mathrm{x}-3)(\mathrm{x}-1)=0 \\ \Rightarrow(x-3)(\mathrm{x}-1)=0 \\ \Rightarrow \mathrm{x}=3,1 \end{array}
So, f^{\prime}(x)>0 if x<1 and x>3 and f^{\prime}(x)<0 if 1<x<3
Therefore, \mathrm{f}(\mathrm{x}) increases on (-\infty, 1) \cup(3, \infty) and \mathrm{f}(\mathrm{x}) is decreasing on interval \mathrm{x} \in(1, 3)

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