Winter Integers

Calculus Level 4

Let $f : \mathbb R \rightarrow \mathbb R$ be a continuous function with

$\large\ { \displaystyle \int _{ 0 }^{ 1 }{ f\left( x \right) f'\left( x \right) dx } = 0 \\ \displaystyle \int _{ 0 }^{ 1 }{ { \left( f\left( x \right) \right) }^{ 2 } } f'\left( x \right) dx = 18 }$ .

Find $\large\ \displaystyle \int _{ 0 }^{ 1 }{ { \left( f\left( x \right) \right) }^{ 4 }f'\left( x \right) dx }$ .

The answer is 97.2.

1 solution

X X
May 26, 2018

$\left.(f(x))^2\right|_0^1=2\int_{ 0 }^{ 1 }{ f\left( x \right) f'\left( x \right) dx } = 0$ $\left.(f(x))^3\right|_0^1=3\int_{ 0 }^{ 1 }{ (f\left( x \right))^2 f'\left( x \right) dx } = 54$ $f^2(1)=f^2(0),f^3(1)-f^3(0)=54\rightarrow f(1)=-f(0),2f^3(1)=54,f(1)=3$ $\left.(f(x))^5\right|_0^1=5\int_{ 0 }^{ 1 }{ (f\left( x \right))^4 f'\left( x \right) dx } = 2\times3^5=486$ $\int _{ 0 }^{ 1 }{ { \left( f\left( x \right) \right) }^{ 4 }f'\left( x \right) dx } =\frac{486}5$

Winter Integers

The answer is 97.2.

1 solution

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