Integral Calculus
Integral of function and its inverse; intersection property
MMTS_Full_Test_21
Grade 12
Question:
A strictly increasing continuous function $f(x)$ intersects its inverse $f^{-1}(x)$ at $x=\alpha$ and $x=\beta$, $\displaystyle\int_\alpha^\beta (f(x)+f^{-1}(x))\,dx=13$, where $\alpha,\beta\in\mathbb{N}$. Then $|\alpha\beta|$ equals
(A) 25
(B) 36
(C) 42
(D) 56
Step-by-Step Solution
Key Concept: For a strictly increasing function and its inverse, both cross the line $y=x$ at intersection. The integral $\int_\alpha^\beta (f(x)+f^{-1}(x))dx = \beta^2-\alpha^2$.
$\int_\alpha^\beta(f(x)+f^{-1}(x))dx=\beta^2-\alpha^2=13$. $\alpha=6$, $\beta=7$, $|\alpha\beta|=42$.
Correct Answer: (C) 42