A calculus problem by Abhinav Jha

Calculus Level pending

If F ( x ) F(x) is a function satisfying F ( x + a ) + F ( x ) = 0 F(x+a) + F(x) = 0 for all x ∈ R x\in \mathbb R and a constant a a such that ∫ b c + b F ( x ) d x \displaystyle \int_b^{c+b} F(x) \, dx is independent of b b , then find the least positive value of c c .

3 a 3a a a 1 a \frac1a 2 b 2b 2 a 2a a + b a+b

Abhinav Jha by Abhinav Jha , 24, India

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1 solution

Abhinav Jha
Oct 13, 2016

F(x+a) +F(x) = 0 for all R ..............1. put 'x+a' in place of 'x' we get a new equation, subtract equation-1 from this new equation. we get F(x + 2a) = F(x) therefore period of the function is '2a' and the given integral is independent of 'b'. So minimum value of 'c' is equal to the period of F(x). ∗ ∗ c r e d i t s ∗ ∗ **credits** : integral calculus. AMIT M AGARWAL

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