Corollaries of the Mean Value Theorem

Definition

The Corollaries of the Mean Value Theorem (MVT) are logical consequences that extend the theorem’s reach, specifically regarding the relationship between a function’s derivative and its overall behavior.

Why It Matters

If the rate of change is zero everywhere, the function must be flat. While it sounds obvious, this corollary is what allows us to prove that two functions with the same derivative differ only by a constant—the fundamental basis for finding antiderivatives-definition.

Core Concepts

• The Constant Function Corollary: If $f'(x) = 0$ for all $x$ in an interval $(a, b)$, then $f$ is constant on $(a, b)$.

  • How to read: “If the derivative of f with respect to x is zero for all x in the interval, then the function f is a constant.”

  • Meaning: Zero derivative everywhere means no change in output—function is flat.

  • Proof logic: MVT states $f(x_2) - f(x_1) = f'(c)(x_2 - x_1)$. If $f'(c) = 0$, then $f(x_2) = f(x_1)$ for any two points, meaning the output never changes.

• The Difference of Constants Corollary: If $f'(x) = g'(x)$ for all $x$ in an interval $(a, b)$, then $f(x) = g(x) + C$ for some constant $C$. This means that functions with identical derivatives are “parallel” versions of each other, differing only by a vertical shift.

  • How to read: “If the derivative of f is equal to the derivative of g everywhere, then f of x is equal to g of x plus a constant C.”
  • Meaning / when to use: Same rate of change implies functions differ only by a constant—justifies the $+C$ in integration.

Connected Concepts

• The Mean Value Theorem
• Antiderivatives
• Fundamental Theorem of calculus, Part 1
• Uniqueness Of Antiderivatives
• Integration Constant
• Path Independence Test
• Fundamental Theorem of calculus
• Scale
• First Principles Thinking