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  • Please solve RD Sharma class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 21 maths textbook solution

Please solve RD Sharma class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 21 maths textbook solution

Answers (1)

Answer:

f(x) is an increasing on R.

Given:

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

To prove:

We have to prove that f(x) is an increasing on R.

Hint:

Show f’(x) > 0 for increasing function.

Solution:

Given

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

On differentiating both sides w.r.t x we get

\begin{aligned} &\Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(x^{3}-6 x^{2}+12 x-18\right) \\ &\Rightarrow f^{\prime}(x)=3 x^{2}-12 x+12 \\ &\Rightarrow f^{\prime}(x)=3\left(x^{2}-4 x+4\right) \\ &\Rightarrow f^{\prime}(x)=3(x-2)^{2} \\ \end{aligned}

\begin{aligned} &\text { As given, } x \in R \\ &\Rightarrow(x-2)^{2}>0 \\ &\Rightarrow 3(x-2)^{2}>0 \\ &\Rightarrow f^{\prime}(x)>0 \end{aligned}

Thus f(x) is increasing function x \in R

Posted by

Gurleen Kaur

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