Given,

y = λ x² + μx + 12

Now, differentiating both sides w.r.t x –

∴ )

Using algebra of derivatives –

⇒

Use:

∴

Now, we have –

f’(x) = 2λx + μ

Given,

f’(4) = 15

⇒ 2λ (4) + μ = 15

⇒ 8λ + μ = 15 ……equation 1

Also f’(2) = 11

⇒ 2λ(2) + μ = 11

⇒ 4λ + μ = 11 …..equation 2

Subtracting equation 2 from equation 1, we have –

4λ = 15 – 11 = 4

∴ λ = 1

Putting λ = 1 in equation 2

4 + μ = 11

∴ μ = 7

Hence,

λ = 1 & μ = 7