Q.Show that the function given by $f(x) = 3x + 17$ is increasing on $\mathbf{R}$.

The key idea is the Increasing Function Test: if $f'(x) > 0$ for all $x$ in an interval, then $f$ is strictly increasing on that interval.

1. Compute the derivative: $f'(x) = \frac{d}{dx}(3x + 17) = 3$.
2. Since $3 > 0$, we have $f'(x) > 0$ for every real number $x$.
3. Therefore, by the Increasing Function Test, $f$ is strictly increasing on $\mathbf{R}$.

The Increasing Function Test is the cleanest way to decide monotonicity for differentiable functions. The idea is simple: the derivative $f'(x)$ tells you the slope of the tangent at $x$. If that slope is positive (or at least non-negative) at every point, the function never goes downhill — it only rises or stays flat. For a strictly increasing function, we need $f'(x) > 0$ everywhere.

Here, $f(x) = 3x + 17$ is a straight line. Its slope is constant, so checking monotonicity is almost trivial — but the derivative method works for any differentiable function, not just lines.

1. Compute the derivative.

   $f'(x) = \frac{d}{dx}(3x + 17) = 3$.

   No $x$ appears — the derivative is the constant $3$.

2. Check the sign.

   $3 > 0$ for every real number $x$. There is no point where the derivative is zero or negative.

3. Apply the Increasing Function Test.

   If $f'(x) \geq 0$ for all $x$ in an interval and $f'(x) > 0$ on any subinterval, then $f$ is increasing on that interval. If $f'(x) > 0$ everywhere, $f$ is strictly increasing.

   Since $f'(x) = 3 > 0$ for all $x \in \mathbf{R}$, the function is strictly increasing on the entire real line.

Method: Proving a Function is Increasing on Its Entire Domain

When a question asks you to show a function is increasing "on $\mathbf{R}$" (or on its whole domain, with no sub-intervals to find), the technique is the Increasing Function Test applied globally rather than piecewise.

Steps

Step 1: Differentiate the function

Find $f'(x)$ using the standard differentiation rules.

Step 2: Determine the sign of $f'(x)$ for every real $x$

Check whether $f'(x)$ is positive for all $x$, without needing to solve any inequality or locate critical points — this happens when $f'(x)$ works out to be a positive constant, or an expression that is manifestly always positive.

Step 3: Invoke the Increasing Function Test

Since the interval here is all of $\mathbf{R}$, one sign check settles the whole domain at once — no sign chart or critical-point analysis is needed.

Step 4 (Applying to this problem): State the conclusion

Conclude that the function is strictly increasing on $\mathbf{R}$, citing the constant sign of the derivative as the justification.

This "one global sign check" shortcut only works when $f'(x)$ never changes sign across the whole domain — for a function whose derivative changes sign, the interval-by-interval sign-chart method is needed instead.