MATH 118g Lecture 8: Functions of Two Variables, Partial Derivatives, and Extrema (sections 8.1-8.6)

EXAMPLE 2 Find the local minimum and maximum values of the function below #x)-3x4-8x3 - 90x* + 7 Video Example SOLUTION f(x) 12x3 - 24x2 -180x 12x(x - 5)(x 3) From the chart Interval 12xx -5 x+3 f(x) decreasing on (-co,-3) +increasing on (-3, 0) decreasing on (0, 5) +increasing on (5, co) x> 5 we see that f(x) changes from negative to posiive at -3, so f(-3) Similarly, f(x) changes from negative to positive at 5, so f(5)- And, f(0) - is a local minimum value by the First Derivative Test. is a local minimum value is a local maximum value because f(x) changes from positive to negative at 0

1. 15 points SCalcET8 4.3.003 My Notes Ask Your Teacher Suppose you are given a formula for a function f (a) How do you determine where f is increasing or decreasing? If f(x) | ? 위 0 on an interval, then f is increasing on that interval. If f'(x) 73) 0 on an interval, then f is decreasing on that interval. (b) How do you determine where the graph of f is concave upward or concave downward? I x) o for all x in I, then the graph of f is concave upward on I. If f"(x) ? 0 for all xin 1, then the graph of f is concave downward on 1. (c) How do you locate inflection points? At any value of x where the concavity changes, we have an inflection point at (x, x)) At any value of x where the concavity does not change, we have an inflection point at (x, fx)) At any value of x where f'(x) = 0, we have an inflection point at (x, f(x)). At any value of x where the function changes from decreasing to increasing, we have an inflection point at (x, (x) At any value of x where the function changes from increasing to decreasing, we have an inflection point at (x, x) Need Help? Rdt Watch It Talk to a Tutor

1,6) G,5) (4,3) -1