Functions
Local extrema of piecewise function with GIF
MJAT_TS6_P1
Grade 12
Question:
Let $f:\mathbb{R}\to\mathbb{R}$ be a real valued function such that $f(x)=|x^2-1|+|x^2-3x+2|-[x^3+x]$ (where $[\cdot]$ denotes GIF). Let $m$ be the number of points of local minima and $M$ be the number of points of local maxima of $f(x)$ in $(0,2)$. Then $m-M$ equals:
Step-by-Step Solution
Key Concept: Analyse $f(x)$ on the intervals $(0,1)$, $(1,2)$ by splitting at the critical points of the absolute value terms and GIF jump discontinuities. $|x^2-1|$ has critical point at $x=1$; $|x^2-3x+2|=|(x-1)(x-2)|$ at $x=1,2$; the GIF introduces jumps at integers.
$m-M=\mathbf{8}$.
Correct Answer: 8