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

Correct Answer: 5

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