Suppose that f and g are continuous on [a, b] and...

Suppose that $f$ and $g$ are continuous on $[a, b]$ and differentiable on $(a, b) .$ Suppose also that $f(a)=q(a)$ and $f^{\prime}(x)<g^{\prime}(x)$ for $a<x<b .$ Prove that $f(b)<g(b) .$ IHint. $[$Hint .
Apply the Mean Value Theorem to the function $h=f-g . ]$

Key Concepts

Continuity on Closed Intervals

This concept ensures that a function has no 'jumps' or 'gaps' within a closed interval, making it possible to apply various analytical tools such as the Extreme Value Theorem and Mean Value Theorem. In problems like this, the continuity on [a, b] is critical because it guarantees that the function behaves well over the entire interval, and limits as the endpoints are reached exist and are meaningful.

Differentiability on Open Intervals

Differentiability means that a function has a well-defined tangent (instantaneous rate of change) at every point in an open interval. In this context, the differentiability of functions on (a, b) is essential for applying the Mean Value Theorem, which relies on the existence of derivatives to relate the overall change in the function values across the interval to the behavior of the derivative at some point.

Mean Value Theorem

The Mean Value Theorem is a fundamental result in calculus that states that for any function that is continuous on [a, b] and differentiable on (a, b), there exists some c in (a, b) such that the derivative at c is equal to the average rate of change over [a, b]. This theorem is used to link local derivative information to a global understanding of the function’s behavior, and it is a key tool in proving inequalities involving function values given inequalities on their derivatives.

Derivative Comparison Principle

This principle involves comparing the derivatives of two functions to infer the relationship between the functions themselves over an interval. If one function’s derivative is consistently less than another’s over an interval, and they share a common value at one endpoint, then the function with the smaller derivative will remain below the other function throughout the interval. This idea is central to proving that f(b) is less than g(b) given that f'(x) < g'(x) and f(a) = g(a).

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