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Question
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
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Solution
\[f'\left( x \right) = \lambda\frac{d}{dx}\left( x^2 \right) + \mu\frac{d}{dx}\left( x \right) + \frac{d}{dx}\left( 12 \right)\]
\[f'\left( x \right) = 2\lambda x + \mu \left( 1 \right)\]
\[\text{ Given }:\]
\[f'\left( 4 \right) = 15\]
\[2\lambda\left( 4 \right) + \mu = 15 \left( \text{ From } \left( 1 \right) \right)\]
\[ \Rightarrow 8\lambda + \mu = 15 \left( 2 \right)\]
\[\text{ Also, given }:\]
\[f'\left( 2 \right) = 11\]
\[2\lambda\left( 2 \right) + \mu = 11 \left( \text{ From } \left( 1 \right) \right)\]
\[4\lambda + \mu = 11 \left( 3 \right)\]
\[\text{ Subtracting equation (3) from equation } (2):\]
\[4\lambda = 4\]
\[\lambda = 1\]
\[\text{ Substituting this in equation } (3):\]
\[4\left( 1 \right) + \mu = 11\]
\[\mu = 7\]
\[\therefore \lambda=1 \text{ and } \mu=7\]
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