Question

Verify Lagrange's mean value theorem for the function f(x)= x⅔ in [-8, 8]
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Answer 1

Answer:

Given, f(x)=x2−x,x∈[1,4]

Since x2−x is continuous on R

and x2−x is exists in [1,4]

⇒f(x) is continuous in [1,4].

Differentiating the given function w.r.t. x

f′(x)=21(x2−x)−1/2.(2x−1)=2x2−x2x−1

which exists ∀x∈R.

∴f(x) is differentiable in (1,4).

Thus, both the conditions of Lagrange's mean value theorem is satisfied therefore, ∃c in (1,4).

Such that f′(c)=4−1f(4)−f(1)

2c2−c2c−1=312

3(2c−1)=2c2−c.

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