Functions
Piecewise function defined recursively — g(x) zeros and limit
MJAT_TS7_P2
Grade 12
Question:
Let $f:[1,\infty)\to\mathbb{R}$ be piecewise defined (alternating linear pieces). Define $g(x)=\int_1^x f(t)\,dt$, $x>1$. Let $\alpha$ = number of solutions of $g(x)=0$ in $(1,8]$, and $\beta=\lim_{x\to 1^+}g(x)/(x-1)$. Then $\alpha+\beta$ equals:
Step-by-Step Solution
Key Concept: The piecewise $f$ is defined on odd and even integer intervals. $\beta=\lim_{x\to 1^+}g(x)/(x-1)=f(1)$ (by L'Hôpital or definition of derivative). Count sign changes of $g$ in $(1,8]$.
$\alpha+\beta=\mathbf{5}$.
Correct Answer: 5