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Question
Verify Lagrange’s mean value theorem for the function f(x) = `sqrt(x + 4)` on the interval [0, 5].
Solution
Given that f(x) = `sqrt(x + 4)` ...(I)
The function f(x) is continuous on the closed interval [0, 5] and differentiable on the open interval (0, 5), so the LMVT is applicable to the function.
Differentiate (I) w. r. t. x.
f'(x) = `1/(2sqrt(x + 4))` ...(II)
Let a = 0 and b = 5
From (I),
f(a) = f(0) = `sqrt(0 + 4)` = 2
f(b) = f(5) = `sqrt(5 + 4)` = 3
Let c ∈ (0, 5) such that
f'(c) = `(f(b) - f(a))/(b - a)`
`1/(2sqrt(c + 4)) = (3 - 2)/(5 - 0) = 1/5`
∴ `sqrt(c + 4) = 5/2`
`\implies` c + 4 = `25/4`
∴ c = `9/4 ∈ (0, 5)`
Thus Lagrange's Mean Value Theorem is verified.
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