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Please solve RD Sharma class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 32 maths textbook solution

Answers (1)

Answer:

f(x) is strictly increasing on R.

Given:

f(x)=x^{3}-3x^{2}+4

To prove:

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

Hint:

For f(x) to be increasing we must have f’(x) > 0.

Solution:

Here we have

f(x)=x^{3}-3x^{2}+4

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

\begin{aligned} &\Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(x^{3}-3 x^{2}+4 x\right) \\ &\Rightarrow f^{\prime}(x)=3 x^{2}-6 x+4 \\ &\Rightarrow f^{\prime}(x)=3\left(x^{2}-2 x+1\right)+1 \\ &f^{\prime}(x)=3(x-1)^{2}+1 \\ &\text { Here } 3(x-1)^{2}+1>0 \text { for all } x \in R \end{aligned}

Hence f(x) is strictly increasing function on R.

Posted by

Gurleen Kaur

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