What are the minimum and maximum values?

Find the absolute maximum and minimum values of the function f(x)=x^4−4x^2+8 for the intervals:?

• Find the absolute maximum and absolute minimum values of the function f(x)=x^4−4x^2+8 on each of the indicated intervals: (A) Interval = [−3,−1]. Absolute maximum = 53 Absolute ...show more

• Answer:

  Well, let's make a brief study of f, considered on R : f(x)=x^4−4x^2+8 -------------------> function is even : x = 0 is symmetry axis then f '(x) = 4x^3 - 8x = 4x(x^2 - 2) = extrema are : x = 0 and x = 2 changement of sign study of f ' gives the increasing / decreasing intervals: f ''(x) = 12x^2 - 8 ---------------------> f ''(0) = -8 < 0 => x = 0 is a max ; -----------> f ''( - √2) > 0 and f ''(√2) > 0 => x = - √2 and x = √2 are local minima lim (x ---> +oo) f(x) = lim (x --> -oo) f(x) = +oo finally : the variations are : x : from -oo to +oo x: -oo _____NEGATIVE________ -√2 ____ POS____ 0 ____NEG___ √2 ____POS ____+oo f(x):+oo_____________________f(-√2) = 4 ________f(0) = 8 _______f(√2) = 4 _________+oo f(x)=x^4−4x^2+8 (A) Interval = [−3,−1] : f is strictly decresaing Absolute maximum = 53 -----> f( -3) = 81 - 36 + 8 = 53 Absolute minimum is of course : f( -1) = 1 - 4 + 8 = 5 (B) Interval = [−4,1] : Absolute maximum = 200 ---> could only be : f(-4) or f(0) : f(-4) = 256-64+8=200 Absolute minimum : can only be: f(-√2) = 4 or f(1) = 1-4+8 = 5 ----> Absolute min = 4 (C) Interval = [−3,4]. Absolute maximum : can only be f(0)=8 or f(4) = 200 ; and not f(-3) (due to variation study) therefore : Absolute MAXimum = 200 Absolute minimum = 4 ---> obvious : f is even and before and after -√2 and √2 , f is decr. and then increasing hope it' ll help !!