Section 4.1 2. (16 pts) Consider the function f(x) =3x2/3 -2x. (a) Explain whether f has absolute extrema on the interval (-co, co). (b) Explain whether f has absolute extrema on the interval [-1, 8].(c) Find the absolute extreme values of f on the interval [-1, 8].Note c is the location of our Definition extreme values, so cis an x-value Let c be a number in the domain D of a function f. . We say that f(c) is an absolute minimum (or global minimum) value if, for all r in D, f( x ) > f (C ) ( c, fcc ) ) . We say that f(c) is an absolute maximum (or global maximum) value if, for all r in D, f (x) =f(c) (ofcc ) ) The minimum and maximum values of f are known as extreme values or just extrema.The Extreme Value Theorem (EVT ) 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) at some numbers c and d in a, b]. we need two hypotheses to hold in order to apply EVT Note: O f is continuous on [a, b] 2 f is defined on a closed interval [aib] then we get f achieves both an abs max and an abs min on that interval [amb]Where are the possible locations that the absolute extrema of a function can occur? has both an alosocute max @endpoints of and an absolute the domain minimum at endpoints of 2) when the its domain. derivative is zero 4 = f ( x ) = 1X1 3). when the denvative does has an absolute min not exist 92020 Kelli Karcher, Daniel KimThe Closed Interval Method To find the absolute extrema of a continuous function on a closed interval. 1. Find the values of fat its critical numbers (if it has critical numbers) 2. Find the values of fat its endpoints 3. The largest of the values from Step 1 and 2 is the absolute maximum value. 4. The smallest of the values from Step 1 and 2 is the absolute minimum value