4.5 Derivatives and the Shape of a Graph

Derivatives reveal a function's behavior, showing where it increases, decreases, or levels off. By analyzing the first and second derivatives, you can understand a graph's shape, including its peaks, valleys, and curves.

Applying derivative tests helps classify critical points as maxima or minima. These tools, along with concepts of continuity and differentiability, give you a powerful way to analyze functions and sketch their graphs.

Derivatives and the Shape of a Graph

Relationship of first derivative to graph

The first derivative $f'(x)$ tells you the slope of the tangent line at any point on $f(x)$. That slope directly tells you whether the function is going up, going down, or momentarily flat.

• Positive $f'(x) > 0$: the function is increasing (heading uphill)
• Negative $f'(x) < 0$: the function is decreasing (heading downhill)
• Zero $f'(x) = 0$: the tangent line is horizontal, marking a critical point

This extends to entire intervals. If $f'(x) > 0$ for every $x$ in an interval, the function is strictly increasing on that whole interval. If $f'(x) < 0$ throughout an interval, it's strictly decreasing there.

First derivative test for extrema

Critical points are where $f'(x) = 0$ or where $f'(x)$ is undefined (while $f(x)$ itself still exists). These are the only candidates for local maxima and minima.

To find and classify them:

1. Compute $f'(x)$.
2. Solve $f'(x) = 0$ and identify any points where $f'(x)$ does not exist.
3. For each critical point, check the sign of $f'(x)$ on either side.

The sign change tells you what kind of extremum you have:

• $f'$ changes from positive to negative → local maximum (function rises then falls)
• $f'$ changes from negative to positive → local minimum (function falls then rises)
• No sign change → neither a max nor a min (think of $f(x) = x^3$ at $x = 0$)

Second derivative and concavity

The second derivative $f''(x)$ measures how the slope itself is changing. This determines the concavity of the graph.

• $f''(x) > 0$: the graph is concave up (curves upward, like a bowl). The slope is getting steeper in the positive direction.
• $f''(x) < 0$: the graph is concave down (curves downward, like an upside-down bowl). The slope is decreasing.

An inflection point is where the concavity switches from up to down or down to up. At an inflection point, $f''(x) = 0$ or $f''(x)$ is undefined. But be careful: $f''(x) = 0$ alone doesn't guarantee an inflection point. You need to confirm that the concavity actually changes on either side.

Concavity test over intervals

To find where a function is concave up or concave down across its domain:

1. Compute $f''(x)$.
2. Find where $f''(x) = 0$ or is undefined. These are your candidate inflection points, and they divide the number line into intervals.
3. Pick a test value in each interval and plug it into $f''(x)$.
4. If $f''(\text{test value}) > 0$, the function is concave up on that interval. If $f''(\text{test value}) < 0$, it's concave down.

Function behavior vs. derivatives

Combining first and second derivative information gives you a complete picture of the graph's shape. There are four possible combinations at any point:

Think of it in terms of speed: "concave up while increasing" means the function is speeding up, while "concave down while increasing" means it's slowing down as it approaches a peak.

Applying Derivative Tests

Second derivative test for extrema

The second derivative test is a quicker alternative to the first derivative test for classifying critical points. Instead of checking sign changes on both sides, you just evaluate $f''$ at the critical point.

Given a critical point where $f'(c) = 0$:

• If $f''(c) > 0$ → the graph is concave up at $c$ → local minimum
• If $f''(c) < 0$ → the graph is concave down at $c$ → local maximum
• If $f''(c) = 0$ → the test is inconclusive; fall back to the first derivative test

The second derivative test saves time when $f''(x)$ is easy to compute. But whenever it gives $f''(c) = 0$, you have no choice but to use the first derivative test and check sign changes directly.

Continuity, differentiability, and limits

These three concepts are closely linked:

• Continuity means the function has no breaks, jumps, or holes at a point. It's a prerequisite for differentiability.
• Differentiability means the function has a well-defined derivative (a non-vertical, unique tangent line) at a point. If a function is differentiable at a point, it's automatically continuous there, but the reverse isn't always true. For example, $f(x) = |x|$ is continuous at $x = 0$ but not differentiable there because of the sharp corner.
• The derivative itself is defined as a limit: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. So every derivative calculation relies on limits working properly, which in turn requires continuity.