Isn't the data insufficient?

Calculus Level 4

If $f(x)$ is a function satisfying $f(x+25) + f(x) = 0$ for all $x \in \mathbb R$ such that,

$\int_{b}^{c+b} f(x) \ dx$

is independent of $b$ , then find the least positive value of $c$ .

1 solution

Kushal Bose
Sep 8, 2016

We know :

$\int_{p}^{p+T} f(x) dx=\int_{0}^{T} f(x) dx$ where $T$ is the period of $f(x)$ .This is independent of $p$

Here $f(x+25)=-f(x)$

$f(x+50)=f(x+25+25)=-f(x+25)=f(x)$

So, f(x) has a fundamental period is $50$

So, from the above formula the value of $c$ will be $\boxed{50}$

Exactly! +1!!

Prakhar Bindal - 4 years, 9 months ago