Calc 1 Minimums and Maximums

Increasing/decreasing test

(a) If f'(x) > 0 on an interval, then f is increasing on that interval. (b) If f'(x) < 0 on an interval, then f is decreasing on that interval.

Critical Number

A number c in the domain of f where f'(c)=0 or f'(c) does not exist.

Local Maximum

A point as high or higher than all nearby points. End points are not local maximums.

Local Minimum

A point as low or lower than all nearby points. End points are not local minimums.

Concave Downward

A region on the graph where the curve opens down, f''(x)<0. Upside down bowl

Concave Upward

A region on the graph where the curve opens up, f''(x)>0. Right side up bowl

Rephrasing of Fermat's Theorem

If f has a local maximum or minimum at c, then c is a critical number of f.

Extreme Value Theorem

If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical numbers in the interval or at endpoints of the interval.

Concavity Test

If f''(x) > 0 for all x in I, then the graph of f is concave upward on I. If f''(x) < 0 for all x in I, then the graph of f is concave downward on I.

Absolute Minimum

The Lowest point over the entire domain of a function. There may be other points that are equally low.

Absolute Maximum

The highest point over the entire domain of a function. There may be other points that are equally high.

Closed Interval Method

To find absolute maximum and minimum values of a continuous function f on a closed interval [a,b]: 1. Find the values of f at the critical numbers of f in (a,b) 2. Find the values of f at the endpoints of the interval 3. The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value

Second Derivative Test

if f'(c) = 0 and f''(c) > 0 then minimum; if f'(c) = 0 and f''(c) < 0 then maximum

Rolle's Theorem

if f(x) is continuous on [a,b] and differentiable on (a,b), and if f(a)=f(b), then there is at least one point (x=c) on (a,b) where f'(c)=0. [DON'T INCLUDE END POINTS]

Mean Value Theorem

if f(x) is continuous on [a,b] and differentiable on (a,b), then there is a number c such that f '(c) = [f(b) - f(a)]/(b - a)