Math 151 Workshop 8 (Extreme Value Theorem, Mean Value Theorem) Write or type your answers, clearly indicating the number of the problem (e.g., #1 or #2(a)). Then either create one PDF of your assignment, or take pictures of it, and upload the PDF or pictures to Gradescope. Unless otherwise stated, you should show your work. Some (but not alll) questions require that you write an explanation. If a question asks for an explanation, then write an explanation in in full, English sentences. Do not write your answers in the margins of this page or underneath the questions. Any submission in which the responses are written on this page will receive a zero. Write your work and answers neatly on clean paper (or on clean digital \"paper\"). Problem 1. (a) Find the absolute extrema of f(x) = v/ 2x on the interval [0, 1] and the location(s) at which the absolute extrema occur. (b) Find the critical points of g(x) = 2sinz + x in the interval (m,7), and use calculus to classify each critical point as a local minimum, and local maximum, or neither. 2-1, -l 9' ( 2 0 ) = 0 9 ( 2 ) = 2 sin ( 217) +2TV = > 2 cosoct 1 = 0 COs ( = - 1 2 .: OC = 21I UIT...... 3 3 3 Now ! 9. " ( 2 ) = - 2 sinic Now at ac = 2 IT 3 9 " ( x ) = - 2 sin (215) 9 " ( 21 ) = - 1.73 .: 9 ( x ) is maxcima at xc = 2 TT 3\fProblem 2 : - X - a . ) The sum of square side and the diameter is 12 feet From the diagram : - 12 - V_ O side of square = I feet (.) Diameter of cirde = (12- 2c ) feet Tolal Area: f( xc ) = 202 + T1 ( 12-20 ) 2 . The domain is the set so Locc127 . The graph it follows : ( Area) 200 + 150+ 100 (5. 274 , 63.346 ) 50 0 2 3 4 6 8 9 10 ( feet ) b.) For Minimum area: f' ( x ) = 206 + ( - II) )(2 ( 12-20) ) f' ( x ) = 2TT - II ( 12 - 20 )For critical Points : f1(20 )= 0 20 - II ( 12 - 20 ) = 0 On Solving : XC = 5. 279 feet Also : f " ( x ) = 2 + II (- 1 ) 2 = 0. 429 . Hence , at ac = 5.219 , f ( 2c ) = minima. . . Minimum Area is obtained when i- Square, side = 5.219 feet Diameter of Circle = 6. 781 feet Now 8 Cost of I square foot of Red Paint = $4 Cost of Isquare foot of Green Paint = $10 For a