The Mean Value Theorem requires continuity on a closed interval and differentiability on its interior; it guarantees an interior tangent matching the secant slope. The Extreme Value Theorem needs continuity on a closed bounded interval and guarantees at least one absolute minimum and one absolute maximum. Critical numbers are interior domain points where the derivative is zero or does not exist; they are candidates, not automatically extrema.
Use a sign chart of the first derivative to determine increase and decrease and to classify local extrema by sign changes. Use the second derivative or the monotonicity of the first derivative to determine concavity. Inflection points require a change in concavity at a point on the graph. A zero second derivative alone gives only a candidate; the second derivative test is inconclusive when that derivative is zero.
On a closed interval where the function is continuous, evaluate the function at every interior critical number and at both endpoints. Compare the actual function values to identify the absolute extrema. Points outside the domain are not candidates. On an open or unbounded domain, analyze endpoint limits and justify whether extrema are attained instead of blindly applying the closed-interval procedure.
Translate graph evidence carefully: the sign of the derivative controls the direction of the function; the direction of the derivative controls the concavity of the function. Zeros on a derivative graph are candidates for horizontal tangents. For an implicit curve find horizontal tangents by setting the numerator of the slope to zero with a nonzero denominator, and reverse the roles for vertical-tangent candidates. Verify points on the original curve.
Define the objective and use the constraint to write it in a single variable. State the feasible domain from the context. Differentiate, locate candidates, and justify the global conclusion using endpoint comparison or a suitable derivative argument. Report the requested physical dimensions or value with units and interpret boundary behavior.
