The second derivative and graphs of f, f' and f''

Introduction to the Second Derivative

In your calculus journey so far, you have learned that the first derivative, \(f'(x)\), tells us the gradient (slope) of a function. But what if we differentiate that result again? This brings us to the second derivative, \(f''(x)\). While the first derivative tells us how fast a graph is rising or falling, the second derivative tells us how the slope itself is changing. This helps us understand the "bend" or "curvature" of a graph, which is essential for sketching accurate functions and solving complex problems.

1. What is the Second Derivative?

The second derivative is simply the derivative of the first derivative. If you have a function \(y = f(x)\), the notation for the second derivative can look like this:

• Lagrange’s notation: \(f''(x)\) (pronounced "f double prime of x")
• Leibniz’s notation: \(\frac{d^2y}{dx^2}\)

A Simple Analogy:
Think of a car moving along a road:
- The original function \(f(x)\) is the car's position.
- The first derivative \(f'(x)\) is the car's velocity (how fast position is changing).
- The second derivative \(f''(x)\) is the car's acceleration (how fast velocity is changing).

Key Takeaway: The second derivative measures the rate of change of the gradient.

2. Concavity: The "Shape" of the Curve

The most important job of the second derivative is to describe concavity. Concavity tells us whether a curve is "cupping upwards" like a bowl or "arching downwards" like a dome.

Concave Upwards (\(f''(x) > 0\))

If the second derivative is positive, the gradient is increasing. The graph looks like a smile \(\cup\). Even if the graph is going down, it is "leveling out" to start going up.

Concave Downwards (\(f''(x) < 0\))

If the second derivative is negative, the gradient is decreasing. The graph looks like a frown \(\cap\). Even if the graph is going up, it is "slowing down" to start going down.

Memory Trick:
- \(f''(x) > 0\) is Positive = Smile (Concave Up)
- \(f''(x) < 0\) is Negative = Frown (Concave Down)

3. Points of Inflexion

A point of inflexion is a point on a curve where the concavity changes (from concave up to concave down, or vice versa).

How to find them:
1. Find where \(f''(x) = 0\) (or where \(f''(x)\) is undefined).
2. Crucial Step: You must check that the concavity actually changes at that point! If \(f''(x)\) is positive on one side and negative on the other, it is a point of inflexion.

Two Types of Points of Inflexion:

1. Stationary Point of Inflexion: Where \(f'(x) = 0\) AND \(f''(x) = 0\). The graph momentarily flattens out while changing concavity (like the graph of \(y = x^3\) at \(x = 0\)).
2. Non-stationary Point of Inflexion: Where \(f''(x) = 0\) but \(f'(x) \neq 0\). The graph is still "moving" up or down as it changes its bend.

Note: For more on finding these, see the "Optimization and points of inflexion" chapter.

4. The Relationship Between \(f\), \(f'\), and \(f''\)

Understanding how these three graphs relate is a favorite topic for IB examiners. Use the table below to see the connections:

If \(f(x)\) is... → Then \(f'(x)\) is... → Then \(f''(x)\) is...
Concave Up → Increasing → Positive (\(> 0\))
Concave Down → Decreasing → Negative (\(< 0\))
Point of Inflexion → At a Max/Min value → Zero (\(= 0\))

Quick Tip: If you are looking at a graph of the first derivative \(f'\), the "turning points" (maxima and minima) of that graph are the points of inflexion on the original graph \(f\).

5. Using the Second Derivative Test

Don't worry if you find the "First Derivative Test" (using sign diagrams) a bit tedious. The Second Derivative Test is often a faster way to determine if a stationary point is a maximum or a minimum.

Step-by-Step Process:
1. Find the first derivative \(f'(x)\) and solve \(f'(x) = 0\) to find your critical values.
2. Find the second derivative \(f''(x)\).
3. Plug your critical value into \(f''(x)\):
- If \(f''(x) > 0\), the graph is concave up (smiling), so the point is a Local Minimum.
- If \(f''(x) < 0\), the graph is concave down (frowning), so the point is a Local Maximum.
- If \(f''(x) = 0\), the test is inconclusive (you must go back to the First Derivative Test/sign diagram!).

Example:
Suppose \(f'(2) = 0\) and \(f''(2) = -5\).
Since the second derivative is negative, the graph is "frowning" at \(x = 2\). Therefore, \((2, f(2))\) is a local maximum.

6. Summary & Common Pitfalls

Key Takeaways:
- \(f'(x)\) tells you direction (Increasing/Decreasing).
- \(f''(x)\) tells you concavity (Up/Down).
- Points of inflexion occur when \(f''(x) = 0\) and the sign of \(f''(x)\) changes.

Common Mistakes to Avoid:
- The "Zero" Trap: Just because \(f''(x) = 0\), it does not automatically mean there is a point of inflexion. You must check that the concavity actually changes sign.
- Mixing up Max/Min: Remember that a positive second derivative means a minimum. It feels counter-intuitive, so think of the "smile" shape to keep it straight.
- Graph Confusion: Always check which graph you are looking at in an exam (is it \(f, f',\) or \(f''\)?). Labels are everything!

Did you know?
In economics, the second derivative is used to see if growth is slowing down or accelerating. Even if a company's profits are still increasing (\(f' > 0\)), a negative second derivative (\(f'' < 0\)) might warn them that their growth is losing steam!