Math 151 Max and Min Theorems

Extreme Value Theorem

If f is continuous on a closed interval [a,b], then f attains an absolute maximum value f( c) and an absolute minimum value f(d) where c and d are in the interval [a,b].

Fermat’s Theorem

If f has a local minimum or maximum at c, and if f’( c) exists, then f’(c ) = 0

Critical Number

c in the domain of f(x) is a critical number if f’( c) = 0 or if f’( c) DNE. (If f has a local max/min at c, then c is a critical number of f).

Rolle’s Theorem

If f(x) satisfies the following: f(x) is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there is a c in the interval (a,b) such that f’( c) =0.

The Mean Value Theorem

If f(x) satisfies the following: continuous on [a,b] and differentiable on (a,b), then there is a number c in the interval (a,b) such that f’( c) = f(b) - f(a)/b-a. (The slope of the tangent line at c is equal to the slope of the secant lie from a to b).

Corollary

If f’(x)=g’(x) for all x in an interval (a,b), then f-g is constant on (a,b); that is, f(x) = g(x) +c where c is a constant.

First Derivative Test

If f’(x) changes from positive to negative at c, then f(x) has a local maximum at c. If f’(x) changes from negative to positive at c, then f(x) has a local minimum at c.

Second Derivative Test

f(x) has a local minimum at c if f’( c) = 0 and f’’( c) is greater than zero (indicates concave up). f(x) has a local maximum at c if f’( c) = 0 and f’’( c) is less than zero (indicates concave down). If f’’( c) = 0 or DNE then go back and use 1st derivative test.

Inflection Points (Second Derivative Test)

a point on a curve y = f(x) is called an inflection point if f is continuous there and the curve changes from CU to CD or from CD to CU at p. They occur when f’’(x) = 0 or when it DNE. They are points where the concavity changes sign (2nd Derivative changes sign).

Concavity Test

If f’’(x) is greater than zero on an interval I, then the graph of f is concave upward on I. If f’’(x) is less than zero on an interval I, then the graph of f is concave downward on I.