Question

Find the value of c for which the conclusion of the mean value theorem holds for the function f(x) = log x on the interval [1, 3]

Answer 1

f(x) = log x, x ∈ [1, 3]

f(3) = log 3,

f(1) = log 1 = 0

Also f'(x) = `1/x`,

∴ f'(c) = `1/c`

By Lagrange's mean value theorem,

f'(c) = `(f(3) - f(1))/(3 - 1)`

∴ `1/c = (log3 - 0)/2`

= `1/2log3`

∴ c log 3 = 2,

∴ c = `2/log_e3`

= 2 log₃ e