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Method of Differentiation: Question

JEE Maths question with a full step-by-step solution.

Question
If f(x)=λx2+μx+12f(x) = \lambda x^2+\mu x+12, f′(4)=15f'(4) = 15 and f′(2)=11f'(2) = 11, then λ+μ\lambda+\mu is.
Solution
Answer: 8
Step 1: Set up the system f′(x)=2λx+μf'(x) = 2\lambda x+\mu.
f′(4)=8λ+μ=15⋯(1)f'(4) = 8\lambda+\mu = 15 \quad \cdots (1)
f′(2)=4λ+μ=11⋯(2)f'(2) = 4\lambda+\mu = 11 \quad \cdots (2)
 Step 2: Solve and sum (1)−(2)(1)-(2): 4λ=4  ⟹  λ=14\lambda = 4 \implies \lambda = 1. Then μ=11−4=7\mu = 11-4 = 7.
λ+μ=8\lambda+\mu = 8
 Answer: 8
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