Unit notes

A critical number lies in the function’s domain and has derivative zero or undefined. An endpoint can be an absolute extremum without being an interior critical number. A statement about f′ is a statement about slopes of f, not necessarily about whether f is positive.

The Extreme Value Theorem guarantees at least one absolute maximum and minimum for a function continuous on a closed interval. Without continuity or a closed bounded interval, the conclusion may fail. The theorem guarantees existence, not an easy formula for the locations.

The Mean Value Theorem requires continuity on [a,b] and differentiability on (a,b). It guarantees a point whose instantaneous rate equals the interval’s average rate. Rolle’s theorem is the equal-endpoint-value case. State the hypotheses before the conclusion.

A critical number lies in the function’s domain where the derivative is zero or does not exist. On a closed interval, absolute-extremum candidates include interior critical numbers and endpoints. Evaluate the original function at candidates to compare heights.

A derivative changing from positive to negative gives a local maximum; negative to positive gives a local minimum. A zero derivative with no sign change need not produce an extremum. Endpoints can be absolute extrema even though the derivative is nonzero there.

A function is concave up where its derivative is increasing and concave down where its derivative is decreasing. A positive second derivative supports concave-up behavior; a negative second derivative supports concave-down behavior. Increasing function values and increasing slopes are different claims.

An inflection point requires a change in concavity at a point on the curve. f″ = 0 produces a candidate but is not sufficient. A second derivative can also be undefined at an inflection point, so inspect intervals rather than only solving one equation.

A graph of f′ above the axis means f increases, not that f itself is positive. A rising graph of f′ means f is concave up. Zeros of f′ become critical-point candidates, while sign changes determine local extrema.

A table or graph of f″ describes changes in f′ and concavity of f. Keep track of which function is shown before interpreting a peak. A local maximum of f′ often marks an inflection of f when the required change in slope behavior occurs.

Define the quantity to optimize, express constraints, and reduce the objective to one variable. Determine the physically meaningful domain before differentiating. A stationary solution outside that domain is not an admissible optimum.

Justify the optimum using endpoint comparison, a derivative sign change, concavity over a valid interval, or another appropriate argument. Merely finding f′ = 0 does not prove a maximum. Include units and translate the optimizing input into every quantity requested by the problem.

Find where f′ is zero or undefined and include boundaries separating parts of the domain. Test the sign of f′ on each open interval. Positive derivative means increasing; negative derivative means decreasing. A change from positive to negative supports a local maximum, and negative to positive supports a local minimum. A zero derivative without a sign change does not establish an extremum.

For absolute extrema on a closed interval where f is continuous, evaluate f at the endpoints and all interior critical numbers. Compare the function values, not derivative values. For an open or unbounded domain, the Extreme Value Theorem does not guarantee an attained maximum or minimum; examine limiting behavior and the actual domain. A supremum approached but never reached is not an absolute maximum.

Concave up means slopes of f are increasing, so f′ is increasing and f″ is positive where the usual derivative criterion applies. A decreasing function can still be concave up if its slopes become less negative. An inflection point requires a change in concavity at a point of the graph, not merely a zero of f″.

When given the graph of f′, locate intervals where that graph rises or falls to determine concavity of f. Where f′ crosses zero, inspect sign changes to locate extrema of f. Where f′ itself has a local extremum, there may be an inflection point of f. These are different questions, so label the graph you are reading before reasoning from its shape.

Define the quantity to optimize and the variable you will use. Use constraints to express the objective in one variable, and write the physically meaningful domain. Differentiate, find candidates, and justify which gives the requested optimum using a sign argument, endpoint comparison, or another appropriate test. A calculator’s local peak is not a complete global argument.

For a rectangle with fixed perimeter, one side can be written in terms of the other; substituting that constraint turns area into a quadratic. For a box or cost problem, the domain may have strict endpoints or excluded values. Keep units throughout. If the problem asks for dimensions, report all dimensions and their interpretation rather than only the input at which a derivative vanished.

The Extreme Value Theorem guarantees absolute extrema for a function continuous on a closed bounded interval. The Mean Value Theorem requires continuity on the closed interval and differentiability on its interior, then guarantees a point where instantaneous rate equals average rate. Name and check these conditions before asserting a conclusion.

A theorem’s failure to apply does not prove its conclusion is false. A discontinuous function might still attain extrema; it simply lacks the stated guarantee. This distinction is useful when evaluating true–false claims or counterexamples.

Rolle’s theorem adds equal endpoint values to the Mean Value Theorem conditions, producing a point with zero derivative. The equal endpoints alone are insufficient if a corner or discontinuity violates a needed hypothesis.

Critical numbers occur in the domain where f′ is zero or undefined. They are candidates, not automatic extrema. A change in f′ from positive to negative gives a local maximum; negative to positive gives a local minimum. No sign change means that test does not establish an extremum.

Concavity follows changes in slope: f″ positive indicates f′ increasing, and f″ negative indicates f′ decreasing on an interval. An inflection point requires a change in concavity at an appropriate point on the graph. Solving f″=0 identifies candidates only.

For absolute extrema on a closed interval, compare function values at relevant critical numbers and endpoints. A local maximum may be smaller than an endpoint value. Report both the location and value when asked, keeping f(c) distinct from c.

Identify the quantity to maximize or minimize and write it as a function. Use constraints to reduce variables, then establish a feasible domain from geometry or context. Differentiating a formula without its domain can produce an impossible candidate.

Find candidates and justify why the chosen one gives the requested optimum. A sign change, second-derivative test where applicable, or comparison of endpoint and critical values provides evidence. Merely setting a derivative equal to zero is not a complete argument.

Interpret the solution with units and practical constraints. If the variable must be an integer, compare feasible nearby values rather than reporting a fractional count. If an endpoint is excluded, a best value may be approached without being attained.

A graph of f′ can establish where f increases and decreases by its sign. Zeros of f′ locate candidates for stationary extrema, but sign changes decide the first-derivative test. Peaks of f′ concern changes in f’s slope, not necessarily peaks of f itself.

Concavity of f follows whether f′ rises or falls. If f′ has a local maximum and changes from increasing to decreasing, f can change from concave up to concave down there. The vertical value of f′ need not be zero at that point. Keep the derivative level labeled on every sketch.

Without one function value or an integration constant, derivative information determines f only up to vertical translation. You can still compare many shape features, but cannot assign an absolute height from slopes alone. This is why an initial value is necessary when reconstructing an accumulation.

A complete extremum argument identifies the candidate, gives the relevant derivative sign or comparison, and states the conclusion. For example: f′ changes from positive to negative at c, so f has a local maximum there. “Because f′(c)=0” leaves out the decisive evidence.

For an absolute maximum on a closed interval, list the relevant candidates and compare f-values. A derivative sign chart can also justify global behavior over the interval, but it must cover the full feasible domain. For a theorem argument, name the conditions and where they hold.

Use the least elaborate valid argument. A numerical table of f-values at all candidates may be clearer than an unnecessarily complicated second derivative. The second-derivative test can be inconclusive when f″=0; that is a reason to use another test, not to announce no extremum.