Exercise 3.3

MA111: Prepared by Dr.Archara Pacheenburawana 27
3. Find all critical numbers and the local extreme values of the following functions.
(a) f(x) = x 2 +5x−1
(b) f(x) = x 3 −3x+1
(c) f(x) = x 3 −3x 2 +3x (d) f(x) = x 4 −3x 3 +2
(e) f(x) = x 3/4 −4x 1/4 (f) f(x) = x 3 −2x 2 −4x
(g) f(x) = sinxcosx, [0,2π] (h) f(x) = x+1
x−1
(i) f(x) = x (j) f(x) = 1
x 2 2
+1
(ex +e −x )
(k) f(x) = x 4/3 +4x 1/3 +3x −2/3 (l) f(x) = 2x √ x+1
(m) f(x) = e −x2
(n) f(x) = sinx 2 , [0,π]
4. Find the absolute maximum and absolute minimum values of f on the given interval.
(a) f(x) = 3x 2 −12x+5, [0,3]
(b) f(x) = 2x 3 +3x 2 +4, [−2,1]
(c) f(x) = x 4 −4x 2 +2, [−3,2]
(d) ) f(x) = x 2 − 2 x , [1 2 ,2]
(e) f(x) = x
x 2 +1 , [0,2]
(f) f(x) = sinx+cosx, [0,π/3]
(g) f(x) = xe −x , [0,2]
(h) f(x) = x−3lnx, [1,4]
(i) f(x) = |x−1|, [0,3]
(j) f(x) = 3√ x, [−1,27]
Answer to Exercise 3.3
1. (a) Absolute maximum at b; local maxima at b, e, and r;
absolute minimum at d; local minima at d and s
(b) Absolute maximum at e; local maxima at e and s;
absolute minimum at t; local minima at b, c, d, r, and t
2. (a) Absolute maximum f(4) = 4; absolute minimum f(7) = 0
local maximum f(4) = 4, f(6) = 3; local minimum f(2) = 1, f(5) = 2
(b) Absolute maximum f(7) = 5; absolute minimum f(1) = 0
local maximum f(0) = 2, f(3) = 4, f(5) = 3
local minimum f(1) = 0, f(4) = 2, f(6) = 1