Let f: R R be a function such that f(2-x) = f(2+x) and f(4-x) = f(4+x), for all x R and _0…

Question

Let $f: R 	o R$ be a function such that $f(2-x) = f(2+x)$ and $f(4-x) = f(4+x)$, for all $x \in R$ and $\displaystyle\int_0^2 f(x)\,dx = 5$. Then the value of $\displaystyle\int_{10}^{50} f(x)\,dx$ is:

Accepted Answer

From $f(2-x)=f(2+x)$, replacing $x 	o 2+x$: $f(-x) = f(4+x) = f(4-x)$ (given).

$\Rightarrow f(x) = f(x+4)$, so period is 4.

$\int_{10}^{50} f(x)\,dx = 10\int_{10}^{14} f(x)\,dx = 10[5+5] = 100$

(Using periodicity, each period-4 interval contributes $\int_0^2 f + \int_2^4 f = 5+5=10$, and there are 10 such periods in $[10,50]$.)

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