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  • please solve rd sharma class 12 chapter16 Increasing and Decreasing Functions exercise Multiple choice question , question 4 maths textbook solution

please solve rd sharma class 12 chapter 16 Increasing and Decreasing Functions exercise Multiple choice question , question 4 maths textbook solution

Answers (1)

best_answer

Correct option (b)

Hint: If f(x)  is increasing function {f}'(x)>0

Given: f(x)=2 \log (x-2)-x^{2}+4 x+1

Explanation: It is given that

f(x)=2 \log (x-2)-x^{2}+4 x+1 

Differentiate w.r.t  x

f^{\prime}(x)=\frac{2.1}{x-2}-2 x+4

since f(x)  is increasing function {f}'(x)>0

\begin{aligned} &\Rightarrow \frac{2}{x-2}-2 x+4>0 \\ &\Rightarrow \frac{1}{x-2}-x+2>0 \end{aligned}

\begin{aligned} &\Rightarrow \frac{-\left(x^{2}-4 x-3\right)}{x-2}>0 \\ &\Rightarrow \frac{(x-3)(x-1)}{2-x}>0 \\ &\Rightarrow(x-3)(x-1)>0 \\ &\Rightarrow x>3 \text { or } x>1 \\ &\Rightarrow x \in(-\infty, 1) \cup(-\infty, 3) \end{aligned}

Since the domain of the function is (2,\infty )

So, x\epsilon (2,3)

Thus, the interval of the function is (2,\infty )

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