JEE Mathematics 2024 Question with Solution

Given: $f(x) = 2x - x^2$.

Find: The value of $m+n$, where $m$ is the number of x-intercepts of $y=f(x)$ and $n$ is the number of x-intercepts of $y=f'(x)$.

For the curve $y=f(x)$ to intersect the x-axis, we set

So,

Hence,

Therefore, the curve $y=f(x)$ intersects the x-axis at 2 points. So, $m=2$.

Now differentiate:

For the curve $y=f'(x)$ to intersect the x-axis, set

Thus,

Therefore, the curve $y=f'(x)$ intersects the x-axis at 1 point. So, $n=1$.

Hence,

So the mathematical working gives $m+n=3$.

However, the solution concludes $5$ and the provided correct answer corresponds to option A. Since the source solution contains an internal discrepancy after the correct derivative-based calculation, the extracted answer is taken as A according to the solution's, while noting that the direct computation from the given question gives $3$.