Question

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

Answer 1

$$f^{\prime }\left(x\right)=\lambda \frac{d}{dx}\left(x^2\right)+\mu \frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(12\right)$$

$$f^{\prime }\left(x\right)=2\lambda x+\mu \left(1\right)$$

$$\text{ Given }:$$

$$f^{\prime }\left(4\right)=15$$

$$2\lambda \left(4\right)+\mu =15\left(\text{ From }\left(1\right)\right)$$

$$\Rightarrow 8\lambda +\mu =15\left(2\right)$$

$$\text{ Also, given }:$$

$$f^{\prime }\left(2\right)=11$$

$$2\lambda \left(2\right)+\mu =11\left(\text{ From }\left(1\right)\right)$$

$$4\lambda +\mu =11\left(3\right)$$

$$\text{ Subtracting equation (3) from equation }(2):$$

$$4\lambda =4$$

$$\lambda =1$$

$$\text{ Substituting this in equation }(3):$$

$$4\left(1\right)+\mu =11$$

$$\mu =7$$

$$\therefore \lambda =1\text{ and }\mu =7$$