Mathematical Proof- Demonstrating Function Concavity

What Concavity Actually Means

Concavity tells you how a function curves. That's it. A function is concave down when its slope decreases as x increases. It's concave up when its slope increases as x increases.

Most textbooks make this sound complicated. It isn't. You just need to know what to look for and how to prove it.

The Second Derivative Test: Your Main Tool

To prove concavity, you use the second derivative. Here's the rule:

• If f''(x) > 0 on an interval, the function is concave up there
• If f''(x) < 0 on an interval, the function is concave down there
• If f''(x) = 0, you can't conclude anything—you need more tests

The second derivative measures the rate of change of the first derivative. When that rate is positive, the slope is increasing. When it's negative, the slope is decreasing.

Step-by-Step Proof Method

Step 1: Find the Second Derivative

Take the derivative twice. If f(x) is your original function, find f'(x), then find f''(x).

Step 2: Identify the Test Interval

Determine where you want to prove concavity. Is it everywhere? Just on (0, ∞)? Be specific.

Step 3: Check the Sign of f''(x)

On your interval, determine whether f''(x) stays positive, negative, or changes sign. This is where students mess up—they check isolated points instead of the entire interval.

Step 4: State Your Conclusion

Match your findings to the definition. "Since f''(x) > 0 for all x in (a,b), f(x) is concave up on (a,b)."

Working Example

Let's prove that f(x) = x³ - 3x² + 2 is concave up when x > 2.

First derivative: f'(x) = 3x² - 6x

Second derivative: f''(x) = 6x - 6

Test: For x > 2, we have 6x - 6 > 6(2) - 6 = 6 > 0

Conclusion: f''(x) > 0 when x > 2, so the function is concave up on (2, ∞).

That's the entire proof. No fluff needed.

Another Example: Parabola

Consider f(x) = x². What's the concavity?

f'(x) = 2x
f''(x) = 2

Since f''(x) = 2 > 0 for all x, the parabola is concave up everywhere. This makes sense—a standard parabola curves upward like a bowl.

Critical Points and Inflection Points

Where f''(x) = 0 or doesn't exist, you might have an inflection point. That's where concavity changes.

To verify an inflection point:

• Find where f''(x) = 0 or is undefined
• Check that f''(x) changes sign at that point
• The function must be continuous there

Example: f(x) = x³

f''(x) = 6x. At x = 0, f''(0) = 0.

For x < 0: f''(x) < 0 (concave down)
For x > 0: f''(x) > 0 (concave up)

Sign changes, so x = 0 is an inflection point.

Concave Down vs Concave Up: Quick Reference

Common Mistakes That Blow the Proof

• Testing a single point: Checking f''(1) > 0 doesn't prove concavity at x = 1. It proves concavity on an interval containing 1.
• Ignoring domain restrictions: f''(x) = 1/x³ is positive for negative x and negative for positive x. Know your domain.
• Confusing concavity with monotonicity: Concave up doesn't mean increasing. f(x) = x² is concave up everywhere but decreases on (-∞, 0).
• Forgetting to simplify: If f''(x) = (x-1)(x-2), test the sign of each factor, not the product's value at one point.

How to Write the Proof Properly

Use this template for your proofs:

1. State the function and interval
2. Compute f''(x)
3. Analyze the sign of f''(x) on the interval
4. Conclude based on the second derivative test

Keep it tight. If f''(x) = 6x + 12, then for x > -2, we have 6x + 12 > 0. Therefore, f is concave up on (-2, ∞).

Functions Without Second Derivatives

Some functions don't have second derivatives everywhere. For f(x) = |x|, the second derivative doesn't exist at x = 0. But you can still analyze concavity using the definition:

f(x) = |x| is concave down on (-∞, 0] and concave up on [0, ∞). The inflection point is at x = 0.

When f''(x) fails to exist, fall back to the original definition: check whether the slope is increasing or decreasing.

The Bottom Line

Proving concavity comes down to one thing: finding where f''(x) is positive or negative. That's the entire game.

Find the second derivative. Check its sign on your interval. State the conclusion. Done.