Sets, Relations & Functions
Zeros of f(x) and f'(x) — Counting Intersections with x-axis
nta_pyq_2024_jan
Grade 11
Question:
Let $f(x)=2^x-x^2$, $x\in\mathbb{R}$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y=f'(x)$ intersect the $x$-axis, then the value of $m+n$ is
Step-by-Step Solution
Key Concept: For $m$: solve $2^x=x^2$. Graph $y=2^x$ and $y=x^2$ — they intersect at 3 points ($x<0$, $x=2$, and one more). For $n$: $f'(x)=2^x\ln2-2x=0$; graph $y=2^x\ln2$ and $y=2x$ — they intersect at 2 points.
$y=f(x)$: $2^x=x^2$ has $m=3$ solutions. $y=f'(x)$: $2^x\ln2=2x$ has $n=2$ solutions. $m+n=5$.
Correct Answer: 5