Question

Consider the function f(x) = (8x^3-4x)^3

(a) Find the derivative

(b) Find critical numbers of f. (Hint there are 5 critical numbers) Round your answers to three decimals.

(c) Fill out the sign chart for the derivative below. Please label the axis as appropriate for your critical numbers.

(d) What are the relative max(es) and min() of f?

Answer 1

we have

a)

b)

f'(x) = 0 for the critical point,

hence the critical point are -0.707, -0.408, 0, 0.408, 0.707

c)

d)

the behaviour of f'(x),

	x < -0.707	x = -0.707	-0.707 < x < -0.408	x = -0.408	-0.408 < x< 0	x = 0	0 < x < 0.408	x = 0.408	0.408 < x < 0.707	x = 0.707	x > 0.707
sign	+	0	+	0	-	0	-	0	+	0	+
behaviour	increasing	saddle	increasing	maximum	decreasing	saddle	decreasing	minimum	increasing	saddle	increasing

the local maximum at x = -0.408

the local minimum at x = 0.408