Least positive value of $c$ if $c, k, b$ are in A.P. is:
Step-by-Step Solution
Key Concept: A functional equation $g(x+a) = -g(x)$ implies periodicity with period $2a$.
From the condition $g(x+a) + g(x) = 0$, we get $g(x+2a) = g(x)$, making $g(x)$ periodic with period $2a$. For any $b$ and $c$ in arithmetic progression with common difference $k$, we have $\int_b^{2k} g(t)dt = \int_a^{b+c} g(x)dx$ using periodicity. Since this integral is independent of $b$, the minimum value of $c$ must be $2a$.
Correct Answer: 4