Question

Verify Lagrange’s mean value theorem for the following functions : f(x) = 2x – x², x ∈ [0, 1].

Answer 1

The function f given as f(x) = 2x – x²  is a polynomial function. Hence, it is continuous on [0, 1] and differentiable on (0, 1).

Thus, the function f satisfies the conditions of Lagrange’s mean value theorem.

∴ there exists c ∈ (0, 1) such that

f'(c) = `(f(1) - f(0))/(1 - 0)`                    ...(1)

Now, f(x) = 2x – x²
∴ f(0) = 0 – 0 = 0
and
f(1) = 2(1) – 1² = 1

Also, f'(x) = `d/dx(2x - x^2)`

= 2 x 1 – 2x
= 2 – 2x
∴ f'(c) = 2 – 2c
∴ from (1), 2 – 2c

= `(1 - 0)/(1)`
= 1
∴ 2c = 1

∴ c = `(1)/(2) ∈ (0 , 1)`
Hence, Lagrange’s mean value theorem is verified.