Question

If for f (x) = λ x² + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.

Answer 1

$$\begin{gathered} f\text{'}\left(x\right)=\lambda \frac{d}{dx}\left(x^2\right)+\mu \frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(12\right) \\ f\text{'}\left(x\right)=2\lambda x+\mu \left(1\right) \\ \text{Given:} \\ f\text{'}\left(4\right)=15 \\ 2\lambda \left(4\right)+\mu =15\left(\mathrm{From}\left(1\right)\right) \\ \Rightarrow 8\lambda +\mu =15\left(2\right) \\ \text{Also, given:} \\ f\text{'}\left(2\right)=11 \\ 2\lambda \left(2\right)+\mu =11\left(\mathrm{From}\left(1\right)\right) \\ 4\lambda +\mu =11\left(3\right) \\ \text{Subtracting equation (3) from equation (2):} \\ 4\lambda =4 \\ \lambda =1 \\ \text{Substituting this in equation (3):} \\ 4\left(1\right)+\mu =11 \\ \mu =7 \\ \text{∴ λ=1 and μ=7} \end{gathered}$$