Analytical Guarantees from Derivatives: MVT and Extrema

Mean Value Theorem (MVT)

If f is continuous on [a,b] and differentiable on (a,b), then there exists at least one c in (a,b) such that f'(c) = (f(b)-f(a))/(b-a).

Secant slope

The average slope of a function over an interval [a,b], computed as (f(b)-f(a))/(b-a).

Tangent slope

The instantaneous slope of the function at a point, given by the derivative f'(x).

Average rate of change

The change in function values over an interval divided by the change in input; equals the secant slope on [a,b].

Instantaneous rate of change

The rate of change at a single point; equals the derivative (tangent slope) at that point.

Continuity on [a,b] (MVT hypothesis)

The function has no breaks or jumps on the closed interval [a,b]; required for MVT to apply.

Differentiability on (a,b) (MVT hypothesis)

The derivative exists at every interior point of (a,b); required for MVT to guarantee a matching tangent slope.

MVT guaranteed point c

A number c in (a,b) where the derivative matches the average rate of change: f'(c) = (f(b)-f(a))/(b-a).

Parallel tangent interpretation (MVT)

MVT guarantees there is at least one point where the tangent line is parallel to the secant line connecting (a,f(a)) and (b,f(b)).

Rolle’s Theorem

A special case of MVT: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c in (a,b) with f'(c)=0.

Derivative-positive implies increasing (MVT application)

Using MVT, if f'(x) > 0 on an interval, then f is increasing on that interval (similarly, f'(x) < 0 implies decreasing).

Uniqueness of solutions (MVT application)

If a derivative condition prevents f' from being zero (or enforces strict monotonicity), MVT can be used to show certain equations have at most one solution.

Motion interpretation of MVT

Over a time interval, if position is differentiable, then at some time the instantaneous velocity equals the average velocity on that interval.

Average velocity

For a position function s(t) on [t1,t2], average velocity is (s(t2)-s(t1))/(t2-t1).

Instantaneous velocity

The velocity at an instant; equals the derivative of position: v(t)=s'(t).

Jump discontinuity (why MVT can fail)

If f is not continuous on [a,b] (e.g., it has a jump), MVT cannot be applied because the average slope may not be matched by any tangent slope.

Nondifferentiable point (why MVT can fail)

Corners, cusps, vertical tangents, or other points where f' does not exist in (a,b) prevent using MVT because differentiability on (a,b) is required.

Extreme Value Theorem (EVT)

If f is continuous on a closed interval [a,b], then f attains an absolute maximum value and an absolute minimum value on [a,b].

Absolute maximum

The highest output value f achieves on the entire interval/domain being considered.

Absolute minimum

The lowest output value f achieves on the entire interval/domain being considered.

Local (relative) maximum

A point x=c where f(c) is greater than nearby function values in some open interval around c.

Local (relative) minimum

A point x=c where f(c) is less than nearby function values in some open interval around c.

Absolute extrema

The absolute maximum and absolute minimum values of a function on a specified interval.

Candidates Test (Closed Interval Method)

To find absolute extrema on [a,b]: test endpoints a and b, plus all interior critical points (where f'(x)=0 or f' does not exist but f is defined), then compare the corresponding f-values.

Critical point (critical number)

An interior x-value where f'(x)=0 or where f'(x) does not exist (while f(x) is defined); these are candidates for extrema.