Question

Question 18

Find the absolute maximum and minimum values of the function over the indicated interval, and indicate the x-values at which they occur. f(x)=x2 - 12/2: 10,141 The absolute maximum value is at x = (Use a comma to separate answers as needed.) The absolute minimum value is atx=0 (Use a comma to separate answers as needed.)

Answer 1

Here we first find the critical numbers of the given function and then calculate the functional values at the critical numbers and end points of the given interval then we identify the absolute extrema.
Here we Given function is f(x) = 23-12; (0,14] Stept! : We find the critical numbers by solving f'(x) = 0 So, f'() = (x3-1222) 34?- (12) (2x) 3x²_ 242 Now f (x) = 0 gives 34?_ 24% = 0 = 3x (4-8) = 0 nEO and n = 8 XO Therefore, there exist there exist two critical numbers and which two belongs in the given interval [0,14 ] on Step 2: Now, We calculate the functional values from (1) at the critical numbers and end points of the given interval to, 14] at the At 2-0 At a=8 512 256 from o flow (0) ?- 1260) = 0 from (1) $(8) = (8)3–1268)2 768 (Lowest) x= 0 (end point of 10,14]), f(0) =0 x=14 ( another end point of co, 14 ]); from (1). (14) = (14)3- 12(14) 2744 - 2352 = 392 (Greaterf) At At

So, Greatest value is absolute maximum and the lowest value is absolute minimum. 392 at x=114 The absolute maximum value is The absolute minimum value is -256 x=18