#### Provide solution for RD Sharma maths class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 30 subquestion (i)

f(x) is an increasing on R

Given:

$f(x)=3x^{5}+40x^{3}+240x$

To prove:

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

Hint:

If f’(x) > 0 then f(x) is increasing function.

Solution:

Given

$f(x)=3x^{5}+40x^{3}+240x$

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

\begin{aligned} &\Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(3 x^{5}+40 x^{3}+240 x\right) \\ &\Rightarrow f^{\prime}(x)=15 x^{4}+120 x^{2}+240 \\ &\Rightarrow f^{\prime}(x)=15\left(x^{4}+8 x^{2}+16\right) \\ &\Rightarrow f^{\prime}(x)=15\left(x^{2}+4\right)^{2} \end{aligned}

\begin{aligned} &\text { Now, } x \in R \\ &\Rightarrow\left(x^{2}+4\right)^{2}>0 \\ &\Rightarrow 15\left(x^{2}+4\right)^{2}>0 \\ &\Rightarrow f^{\prime}(x)>0 \end{aligned}

Thus f(x) is increasing function for all x $\in$ R.

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