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

Answer 1

Step 1: Find Points where $y = f(x)$ Intersects the x-axis

For $f(x) = 2^x - x^2 = 0$, we determine the values of $x$ where this equation holds. This equation intersects the x-axis at three points, so $m = 3$.

Step 2: Find Points where $y = f'(x)$ Intersects the x-axis

The derivative $f'(x) = 2^x \ln 2 - 2x$. Setting $f'(x) = 0$, we solve $2^x \ln 2 = 2x$ to find the points where this equation intersects the x-axis. This yields two intersection points, so $n = 2$.

Step 3: Calculate $m + n$

$m + n = 3 + 2 = 5$

So, the correct answer is: 5