Determine whether f satisfies the hypotheses of the mean...

Determine whether $f$ satisfies the hypotheses of the mean value theorem on $\mid a, b],$ and, if so, find all numbers $c$ in $(a, b)$ such that
$$
f(b)-f(a)=f^{\prime}(c)(b-a)
$$
$$
f(x)=3 x^{2}+x-4
$$

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Key Concepts

Differentiability

Differentiability means that a function has a derivative at every point in an open interval. This requirement of the Mean Value Theorem guarantees that the function's graph is smooth (without sharp corners or cusps) on (a, b), so the instantaneous rate of change is well-defined at every point.

Derivative Application

Calculating and applying the derivative is crucial to finding the specific point c that satisfies the conditions of the Mean Value Theorem. It involves differentiating the function, then equating the derivative at c to the average rate of change of the function over the interval, and solving for c.

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental result in calculus which states that for any function that is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one point c in (a, b) such that the instantaneous rate of change (derivative) at c equals the average rate of change over [a, b].

Continuity

Continuity is a property of functions where small changes in the input lead to small changes in the output. For the Mean Value Theorem to apply, the function must be continuous on the entire closed interval [a, b], ensuring there are no breaks or jumps in the graph of the function.

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