Find the function from numbers

Calculus Level 3

Let $f:(1,+\infty)\to \mathbb R$ be a continuous function such that:

$\int_{2}^{150} (x-1)\ln(x-1)(2f(x)-(x-1)\ln(x-1))dx=\int_{2}^{150}f^{2}(x)dx$

If $f(101)=A$ , enter $\lfloor A\rfloor$ as your answer.

Notation: $\lfloor \cdot \rfloor$ denotes the floor function .

The answer is 460.

1 solution

Maximos Stratis
Jun 4, 2017

We can move the LHS integral to the RHS to get:
$\int_{2}^{150}(f^{2}(x)-(x-1)ln(x-1)(2f(x)-(x-1)ln(x-1)))dx=0\Rightarrow$
$\int_{2}^{150}(f^{2}(x)-2f(x)(x-1)ln(x-1)+((x-1)ln(x-1))^{2})dx=0\Rightarrow$
$\int_{2}^{150}(f(x)-(x-1)ln(x-1))^{2}dx=0$
But, $(f(x)-(x-1)ln(x-1))^{2}\geq 0$ , so:
$(f(x)-(x-1)ln(x-1))^{2}=0\Rightarrow$
$f(x)-(x-1)ln(x-1)=0\Rightarrow$
$f(x)=(x-1)ln(x-1)$
Therefor:
$f(101)=100\times ln(100)\Rightarrow$
$A\approx 460.517\Rightarrow$
$\boxed{\lfloor A\rfloor=460}$

You have to mention why the integrand will be zero

Kushal Bose - 4 years ago

Find the function from numbers

The answer is 460.

1 solution

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