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Question
Verify Langrange’s mean value theorem for the function:
f(x) = x (1 – log x) and find the value of c in the interval [1, 2].
Solution
Given function ‘f’ is continuous in [1, 2] and differentiable in (1, 2)
f(x) = x (1 – log x) = x – x log x
f'(x) = 1 – x × – log x = 1 – 1 – log x
f'(x) = – log x
According to Langrange’s Mean Value Theorem, E a real number c ∈ (1, 2) s.t.,
`("f"("b")- "f"("a"))/("b" - "a") = "f"("c")`
`("f"(2) - "f"(1))/(2-1) = - "log" "c"`
2 - 2 log 2 - (1 - lo g1) = - log c
1 - log 4 = - log c
⇒ log 4 - log c = 1
⇒ `"log"_"e"(4/"c") = 1`
⇒ e = `4/"c" => "c" = 4/"e"`
Now , 2 < e < 4 ⇒ `1/2 > 1/"e" > 1/4 =>4/2 > 4/"e" > 4/4 => 2 > 4/"e" > 1`
`therefore "c" = 4/"e" in (1,2)`
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