Given that f and g are continuous on [a, b], that f(a) g(a),...

Given that $f$ and $g$ are continuous on $[a, b],$ that $f(a)$ $g(a),$ and $g(b)<f(b),$ show that there exists at least one number $c$ in $(a, b)$ such that $f(c)=g(c)$. HINT: Consider $f(x)-g(x).$

Key Concepts

Continuity

Continuity is a fundamental property of functions that ensures there are no sudden jumps or breaks in the function's graph. A continuous function on a closed interval [a, b] means that for every point within the interval, the function's value changes smoothly and predictably. This property is crucial when applying various theorems in calculus, such as the Intermediate Value Theorem, as it guarantees that the function takes on every intermediate value between f(a) and f(b).

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on two values f(a) and f(b) at the endpoints, then for any value between f(a) and f(b), there exists at least one point in the interval where the function equals that intermediate value. In proofs like the one given, this theorem is used to show the existence of a point c where the difference of two continuous functions becomes zero, leading to the conclusion that f(c) equals g(c).

Function Transformation via Difference

Transforming the problem by considering the difference of two functions, such as f(x) - g(x), is a common technique in analysis. This approach simplifies the problem by consolidating the equation f(x) = g(x) into finding the root of the function h(x) = f(x) - g(x). The continuity of f and g ensures that h(x) is also continuous, which allows the application of the Intermediate Value Theorem to establish the existence of a solution where h(c) = 0, thereby proving that f(c) equals g(c).

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