Section 4.3 Optimization

Often we want to find the maximum or minimum value of a function This is called optimization Absolute (Global) Maxima and Minima f has an absolute (or global) minimum at p if f(p) is less than or equal to all values of f f has a global maximum at p if f(p) is greater than or equal to all values of f Absolute (or global) maxima and minima are sometimes referred to as “extrema” or “optimal values”

To find the absolute max and mins of a continuous function on a closed interval (i.e. endpoints are included): Compare function values at the critical points and endpoints To find the global maxima and minima of a continuous function on an open interval (i.e. endpoints not included or infinite endpoint): Find the value of the function at all critical points and sketch the graph. Look at the function values when x approaches the endpoints of the interval, or approaches ±∞, when appropriate.

Let’s find the absolute max and min on the following intervals Consider the function It has a critical points at And Let’s find the absolute max and min on the following intervals In which cases can we find bounds for this function and what are they?

Extreme Value Theorem If f is a continuous function on the closed interval a ≤ x ≤ b, then f has a global maximum and a global minimum on that interval If f is continuous over a closed interval, we are guaranteed to have an absolute max and absolute min They either occur at critical points or at the endpoints Thus the procedure is to find all critical points of f, evaluate f at the critical points and endpoints and compare values

Example Find the absolute max and min of on [0, 2] Let’s take a look at #25 in the book