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  • Explain solution rd sharma class 12 chapter 16 Increasing and decreasing function exercise multiple choice question, question 32

Explain solution rd sharma class 12 chapter 16 Increasing and decreasing function exercise multiple choice question, question 32

Answers (1)

 1<x<3, Option (a)

Hint: Take a derivative of given equation

Given: y=x(x-3)^{2} decreases for the values

Solution: We have, y=x(x-3)^{2}

                                    \begin{aligned} &y=x\left(x^{2}-6 x+9\right) \\ &y=x^{3}-6 x^{2}+9 x \end{aligned}

                                     \begin{aligned} \frac{\mathrm{dy}}{\mathrm{dx}} &=3 \mathrm{x}^{2}-12 \mathrm{x}+9 \\ \frac{\mathrm{dy}}{\mathrm{dx}} &=3\left(\mathrm{x}^{2}-4 \mathrm{x}+3\right) \\ &=3(\mathrm{x}-3)(\mathrm{x}-1) \end{aligned}

Y = f(x) decreases when \begin{aligned} \frac{\mathrm{dy}}{\mathrm{dx}}<0 \end{aligned}

The sign scheme of \begin{aligned} \frac{\mathrm{dy}}{\mathrm{dx}} \end{aligned} is shown,

                

               ∴ f’(x) =                                  

From the sign scheme \begin{aligned} \frac{\mathrm{dy}}{\mathrm{dx}}<0 \end{aligned}

               For x\epsilon (1,3)

               y=x(x-3)^{2}  decreases when x\epsilon 1<x<3

 

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