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  • Provide solution for RD Sharma maths class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 18

Provide solution for RD Sharma maths class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 18

Answers (1)

Answer:

f(x)=(x-1)e^{x}+1 \text { is an increasing function for all x }> 0.

Given:

f(x)=(x-1)e^{x}+1

To prove:

\text { We have to show that } f(x)=(x-1)e^{x}+1 \text {is an increasing function for all x}>0.

Hint:

f’(x) > 0 is condition for increasing function of f(x).

Solution:

Given

f(x)=(x-1)e^{x}+1

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

\begin{aligned} &\Rightarrow f^{\prime}(x)=\frac{d}{d x}\left[(x-1) e^{x}+1\right] \\ &\Rightarrow f^{\prime}(x)=e^{x}+(x-1) e^{x} \\ &\Rightarrow f^{\prime}(x)=e^{x}(1+x-1) \\ &\Rightarrow f^{\prime}(x)=x e^{x} \\ &\text { As given } x>0 \\ &\Rightarrow e^{x}>0 \\ &\Rightarrow x e^{x}>0 \\ &\Rightarrow f^{\prime}(x)>0 \end{aligned}

Hence condition of f(x) to be increasing is satisfied.

Thus f(x) is increasing on interval x>0

Posted by

Gurleen Kaur

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