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Derivatives — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSDerivatives4 Marks
Question
If for $\text{f}(\text{x})=\lambda\text{x}^2+\mu\text{x}+12,\text{f}'(\text{x})=15$ and $\text{f}'(\text{2})=11,$ then find $\lambda$and $\mu.$
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Answer

We have,$\text{f}(\text{x})=\lambda\text{x}^2+\mu\text{x}+12$
$\Rightarrow\text{f}'(\text{x})=\text{2x}\lambda+\mu\dots(\text{i})$
but, $\text{f}'(4)=15$
from (i)
$8\lambda+\mu=15\dots(\text{ii})$
also, $\text{f}'(\text{12})=11$
$4\lambda+\mu=11\dots(\text{iii})$
(ii)-(iii) gives
$4\lambda=4$
$\Rightarrow\lambda=1$
from (ii)
$8.1+\mu=15$
$\Rightarrow\mu=7$
Hence,
$\lambda=1$ and $\mu=7$

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