Functions
Polynomial from functional equation — one-one
MJAT_TS8_P1
Grade 12
Question:
Let $f$ be a polynomial on non-negative reals satisfying $f(x)f(y)+2=f(x)+f(y)+f(xy)$, $f$ is one-one, $f(0)=1$, $f'(1)=2$. Which is/are true?
A) $f(5)=26$
B) $h(x)=\min\{f(x)/2,x^2,1-x\}$ ($x\geq 0$) has 3 points of non-differentiability
C) $f(5)=29$
D) $h(x)=\min\{f(x)/2,x^2,1-x\}$ ($x\geq 0$) has 2 points of non-differentiability
Step-by-Step Solution
Key Concept: $(f(x)-1)(f(y)-1)=f(xy)-1$: let $g(x)=f(x)-1$, then $g(xy)=g(x)g(y)$. So $g(x)=x^n$ (power function). $f(0)=1\Rightarrow g(0)=0$. $f'(1)=2\Rightarrow g'(1)=2\Rightarrow n=2$. Thus $f(x)=x^2+1$.
A ✓, B ✓. Answer: A, B.
Correct Answer: AB