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Solution for If the Value of C Prescribed in Rolle'S Theorem for the Function F (X) = 2x (X − 3)N on the Interval [ 0 , 2 √ 3 ] is 3 4 , Write the Value of N (A Positive Integer) ? - CBSE (Commerce) Class 12 - Mathematics

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Question

If the value of c prescribed in Rolle's theorem for the function f (x) = 2x (x − 3)n on the interval \[[0, 2\sqrt{3}] \text { is } \frac{3}{4},\] write the value of n (a positive integer) ?

Solution

We have

\[f\left( x \right) = 2x \left( x - 3 \right)^n\]

Differentiating the given function with respect to x, we get

\[f'\left( x \right) = 2\left[ xn \left( x - 3 \right)^{n - 1} + \left( x - 3 \right)^n \right]\]

\[ \Rightarrow f'\left( x \right) = 2 \left( x - 3 \right)^n \left[ \frac{xn}{\left( x - 3 \right)} + 1 \right]\]

\[ \Rightarrow f'\left( c \right) = 2 \left( c - 3 \right)^n \left[ \frac{cn}{\left( c - 3 \right)} + 1 \right]\]

Given:

\[f'\left( \frac{3}{4} \right) = 0\]

\[\therefore 2 \left( \frac{- 9}{4} \right)^n \left[ \frac{\frac{3}{4}n}{\left( \frac{- 9}{4} \right)} + 1 \right] = 0\]

\[ \Rightarrow 2 \left( \frac{- 9}{4} \right)^n \left[ \frac{- n}{3} + 1 \right] = 0\]

\[ \Rightarrow \left[ \frac{- n}{3} + 1 \right] = 0\]

\[ \Rightarrow - n + 3 = 0\]

\[ \Rightarrow n = 3\]

  Is there an error in this question or solution?
Solution If the Value of C Prescribed in Rolle'S Theorem for the Function F (X) = 2x (X − 3)N on the Interval [ 0 , 2 √ 3 ] is 3 4 , Write the Value of N (A Positive Integer) ? Concept: Maximum and Minimum Values of a Function in a Closed Interval.
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