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].

Answer 1

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)`