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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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