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provide solution for rd sharma maths class 12 chapter 14 Mean Value Theoram exercise  multiple choice question 2

Answers (1)

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Answer:

Option (c)

Hint:

Use Rolle’s Theorem.

Given:

4a+2b+c=0  and 3ax^{2}+2bx+c=0 has at least one real root.

Solution:

Let,    f(x)=ax^{3}+bx^{2}+cx+d

          f(0)=d

            f(2)=a\left ( 2 \right )^{3}+b\left ( 2 \right )^{2}+c\left ( 2 \right )+d

                     =8a+4b+2c+d

                     =2(4a+2b+c)+d

                     =0+d                \left [ \because 4a+2b+c=0 \right ]

           f(2)=d      

  \therefore fis continuous in closed interval [0,2]

            f(0)=f(2)

As per Rolle’s Theorem,

          f^{'}(x)=3ax^{2}+2bx+c

          f^{'}(\alpha )=3a\alpha ^{2}+2b\alpha +c

          3a\alpha ^{2}+2b\alpha +c=0

Hence, equation f(x) has at least one root in the interval (0,2)

\therefore f(x) must have one root in the interval (0,2) .

So, option (c) is correct.

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