Question bank/JEE Main/Mathematics
JEE MainMathematicsModerateFunctions
The values of a, m and b for which Lagrange's mean value
theorem is applicable to the function f(x) for x
$f(x) = \left\{ \begin{matrix}
3 & x = 0 \\
- x^{2} + a & 0 < x < 1 \\
mx + b & 1 \leq x \leq 2
\end{matrix} \right.\ $
Show answer and solution
$f(x) = \left\lbrack \begin{matrix}
3 & x = 0 \\
a - x^{2} & 0 < x < 1 \\
mx + b & 1 \leq x \leq 2
\end{matrix} \right.\ $
For 2]
and differentiable in (0, 2) Hence, a = 3
m + b = 2..........(i)
$f^{'}(x) = \left\lbrack \begin{matrix}
- 2x & 0 < x < 1 \\
m & 1 < x < 2
\end{matrix} \right.\ $
m = - 2
and hence b = 4
3 & x = 0 \\
a - x^{2} & 0 < x < 1 \\
mx + b & 1 \leq x \leq 2
\end{matrix} \right.\ $
For 2]
and differentiable in (0, 2) Hence, a = 3
m + b = 2..........(i)
$f^{'}(x) = \left\lbrack \begin{matrix}
- 2x & 0 < x < 1 \\
m & 1 < x < 2
\end{matrix} \right.\ $
m = - 2
and hence b = 4
Source: Grass-G / M-19 Mathematics Paper 28 Nov Fourth.docx
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