Q.The interval in which the function $f$ defined by $f(x)=e^{x}$ is strictly increasing, is
(A) $[1,\infty)$
(B) $(-\infty,0)$
(C) $(-\infty,\infty)$
(D) $(0,\infty)$

Concept: Monotonicity Of Exponential — The exponential function $e^x$ has a positive derivative everywhere, so it is strictly increasing on its entire domain.

Step 1: Compute the derivative:

$f'(x) = \frac{d}{dx} e^x = e^x$.

Step 2: Since $e^x > 0$ for all real $x$, we have $f'(x) > 0$ for every $x \in \mathbb{R}$. …

The core idea here is monotonicity of the exponential function. A function is strictly increasing on an interval if its derivative is positive at every point in that interval. For $f(x) = e^{x}$, the derivative is $f'(x) = e^{x}$, which is always greater than zero — no matter what real number $x$ you plug in. This means the function never flattens out or decreases; it climbs steadily from left to right across the whole number line.

Let’s walk through the reasoning step by step.

1. Recall the derivative rule. The derivative of $e^{x}$ with respect to $x$ is itself:

This is a fundamental result — no chain rule needed here, just the pure exponential.

2. Check the sign of the derivative. For any real number $x$, the value $e^{x}$ is always positive. Why? Because $e > 0$, and a positive number raised to any real power stays positive. So:

3. Connect derivative sign to monotonicity.

   A function is strictly increasing on an interval if its derivative is positive at every interior point of that interval. Since $f'(x) > 0$ holds for every real $x$, the function is strictly increasing on the entire set of real numbers: $(-\infty, \infty)$.

4. Examine the given options.

   • (A) $[1, \infty)$ — True, but too narrow; the function is increasing everywhere, not just from 1 onward.
   • (B) $(-\infty, 0)$ — Also true, but again only a subset. …

Method: Determining Where a Function Is Strictly Increasing via the Sign of Its Derivative

Use this method for any "find the interval of increase/decrease" question — the derivative's sign is the entire test.

Steps

Step 1: Compute the derivative

Differentiate $f(x)$ using the standard rules.

Step 2: Determine the sign of $f'(x)$ across the relevant domain

Either find the critical points where $f'(x)=0$ or is undefined, and test the sign of $f'(x)$ in each resulting sub-interval, or — when the derivative has a recognizably fixed sign, such as $f'(x)=e^x$, which is positive for every real $x$ — conclude the sign directly without needing to split into sub-intervals.

Step 3: Apply the increasing/decreasing rule …

Common Mistakes

Mistake 1: Picking a subset interval instead of the full interval

A student sees $e^x$ growing very fast for large positive $x$ and assumes the "increasing" behavior only really counts on $(0,\infty)$ or $[1,\infty)$, missing that $e^x$ is climbing (just more slowly) for negative $x$ too. Why it's wrong: strictly increasing only requires $f'(x)>0$, not any particular rate of growth — a slow climb still counts. Correct approach: check the sign of the derivative, not the magnitude of the function's values, and take the full interval where that sign condition holds.

Mistake 2: Confusing "positive values" with "increasing" …