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  • Explain Solution RD Sharma Class 12 Chapter 14 Inverse Trigonometric Function Exercise Very Short Answer Question 4 Maths.

Explain Solution RD Sharma Class 12 Chapter 14 Inverse Trigonometric Function Exercise Very Short Answer Question 4 Maths.

Answers (1)

Answer:

                n=3

Hint:

Find differentiate of f\left ( x \right ) and then apply Rolle’s Theorem.

Given:

                f\left ( x \right )=2x\left ( x-3 \right )^{n},x\: \epsilon \left [ 0,2\sqrt{3} \right ]and value of c is \frac{3}{4}.

Solution:

According to Rolle’s Theorem

                \begin{array}{ll} & f(a)=f(b) \\\\ \Rightarrow & f(0)=f(2 \sqrt{3}) \\\\ \Rightarrow & 0=2 \cdot 2 \sqrt{3}(2 \sqrt{3}-3)^{n} \end{array}

                \Rightarrow \left ( 2\sqrt{3}-3 \right )^{n}=0

Now,     f\left ( x \right )=2x\left ( x-3 \right )^{n}

Differentiate with respect to x,

                {f}'\left ( x \right )=2\left ( x-3 \right )^{n}+2x\cdot n\left ( x-3 \right )^{n-1}                                           \left [ \because \frac{d\left ( uv \right )}{dx}=u\frac{dv}{dx} +v\frac{du}{dx}\right ]

Applying Rolle’s Theorem,

                {f}'\left ( c \right )=2\left ( c-3 \right )^{n}+2c\cdot n\left ( c-3 \right )^{n-1}                                        

                {f}'\left ( c \right )=0          

2\left ( c-3 \right )^{n}+2c\cdot n\left ( c-3 \right )^{n-1}=0

\Rightarrow 2\left ( \frac{3}{4}-3 \right )^{n}+2\cdot \frac{3}{4}\cdot n\left ( \frac{3}{4}-3 \right )^{n-1}=0                                                                                                   \left [ \because c=\frac{3}{4} \right ]

\Rightarrow \left ( \frac{3}{4}-3 \right )^{n}\left [ 2\left ( \frac{3}{4}-3 \right )+\frac{3n}{2}\right ]=0        

\Rightarrow \left ( \frac{3}{4}-3 \right )^{n-1}=0          or  2\left ( \frac{3-12}{4} \right )+\frac{3n}{2}=0

\Rightarrow                                                \frac{3-12+3n}{2}=0

                                                     3-12+3n=0

                                                     3n=9

                                                     n=3

Posted by

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