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Provide solution for RD Sharma maths class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 10

Answers (1)

Answer:

f(x) is increasing for all x \in R

Given:

f(x)=x^{3}-15x^{2}+75x-50

To prove:

We have to show that f(x) is increasing for all x \in R

Hint:

Show f’(x)>0 for f(x) to be increasing.

Solution:

Given

f(x)=x^{3}-15x^{2}+75x-50

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

\begin{aligned} &\Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(x^{3}-15 x^{2}+75 x-50\right) \\ &\Rightarrow f^{\prime}(x)=3 x^{2}-30 x+75 \\ &\Rightarrow f^{\prime}(x)=3\left(x^{2}-10 x+25\right) \\ &\Rightarrow f^{\prime}(x)=3(x-5)^{2} \end{aligned}

Now, as given x \in R

\begin{aligned} &\Rightarrow(x-5)^{2}>0 \\ &\Rightarrow 3(x-5)^{2}>0 \\ &\Rightarrow f^{\prime}(x)>0 \end{aligned}

Hence condition for f(x) to be increasing.

Thus f(x) is increasing on interval x \in R.

Posted by

Gurleen Kaur

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