Functions
Median function — continuity and differentiability
MJAT_TS3_P1
Grade 12
Question:
If $P(x)=\mathrm{mid}(g(x),h(x),f(x))$ means the median value of the three functions at each $x$. Let $P(x)=\mathrm{mid}\left(x-1,\;(x-3)^2,\;3-\dfrac{(x-2)^2}{3}\right)$ for $x\in[1,3]$. Then:
A) Number of points of discontinuity of $P(x)$ is $0$
B) Number of points of discontinuity of $P(x)$ is $1$
C) Number of points of non-differentiability of $P(x)$ is $1$
D) Number of points of non-differentiability of $P(x)$ is $2$
Step-by-Step Solution
Key Concept: Find where the ordering of $f_1=x-1$, $f_2=(x-3)^2$, $f_3=3-(x-2)^2/3$ changes on $[1,3]$. The median function switches between the three functions at crossover points. Since all three functions are continuous, $P(x)$ is continuous (A ✓, B ✗).
All component functions are continuous so $P$ is continuous: A ✓. Non-differentiability: solve $f_1=f_2$, $f_2=f_3$, $f_1=f_3$ for crossovers — 2 points in $(1,3)$. Answer: A, D.
Correct Answer: AD