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

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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