Question

If for $f\left(x\right)=\lambda x^2+\mu x+12,f^{\prime }\left(4\right)=15$ and $f^{\prime }\left(2\right)=11$ then find $\lambda$ and $\mu .$

Answer 1

$f\left(x\right)=\lambda x^2+\mu x+12$

$f\left(x\right)=2\lambda x+\mu$

$f\left(4\right)=2x\lambda +\mu =15$ -----(i)

$f\left(2\right)=4\lambda +\mu =11$ ------(ii)

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$4\lambda =4$ substituting form eq (i) to eq (ii)

$\lambda =1$

putting volume $\lambda =1$ eq (i)

$8+\mu =15$

$\mu =7$

$\lambda =1$