GATE CS 2017 Set 2 QGA-9 — General Aptitude

Let $f(x) = e^x + 0.5x^2 - 2$. To find the number of roots in the interval $[-5, 5]$, we analyze the behavior of $f(x)$ using its derivative:

To find the critical points, we set $f'(x) = 0$:

Since $e^x$ is strictly increasing and $-x$ is strictly decreasing, there is exactly one real solution, say $x_0$, where $f'(x_0) = 0$. Since $f'(-1) = e^{-1} - 1 < 0$ and $f'(0) = 1 > 0$, this critical point lies in the interval $(-1, 0)$.

We analyze the behavior of $f(x)$ around $x_0$:

• For $x < x_0$, $f'(x) < 0$, so $f(x)$ is strictly decreasing.
• For $x > x_0$, $f'(x) > 0$, so $f(x)$ is strictly increasing.

Thus, $f(x)$ has a unique global minimum at $x_0$.

Now, let's evaluate $f(x)$ at the boundaries and key points:

• At $x = -5$: $f(-5) = e^{-5} + 12.5 - 2 \approx 10.5 > 0$
• At $x = 0$: $f(0) = e^0 + 0 - 2 = -1 < 0$ (which implies the minimum value $f(x_0) < f(0) < 0$)
• At $x = 5$: $f(5) = e^5 + 12.5 - 2 \approx 158.9 > 0$

By the Intermediate Value Theorem:

1. Since $f(-5) > 0$ and $f(x_0) < 0$, there is exactly one root in $[-5, x_0]$.
2. Since $f(x_0) < 0$ and $f(5) > 0$, there is exactly one root in $[x_0, 5]$.

Thus, the function has exactly 2 roots in the interval $[-5, 5]$.