Let fx x3 4x Determine on which intervals fx greaterthanor

Let f(x) = x^3 - 4x. Determine on which intervals f(x) greaterthanorequalto 0. Let g(x) = 2x - 3. Evaluate (f compositefunction g) (1) and (g compositefunction f) (0). Evaluate f(x + h) - f(x)/h. Simplify as much as possible (which should result in the elimination of h from the denominator).

Solution

f(x) = x^3 -4x

a) f(x) >=0

x^3 -4x >0 =0 ; x(x^2 -4) >=0

x(x+2)(x-2) >=0

-2<=x<= 0 or x >=2

[ -2, 0] U [ 2, inf )

b) g(x) = 2x -3   ;

(f o g)(x) = (2x-3)^3 -4(2x -3)

(f o g)(1) = (2-3)^3 - 4(2-3) = -1 +4 = 3

(g o f)(x) = 2(x^3 - 4x) -3

(g o f)(1) = 2( 1- 4) -3 = -9

c) f(x+h) - f(x) = ( x+h)^3 - 4(x+h) - x^3 + 4x

= (x +h -x)( x^2 +h^2 +2hx + x^2 +hx +x^2) -4h

= h( 2x^2 + 3hx +x^2) -4h

[f(x+h) -f(x)]/h = 2x^2 + 3hx +x^2 -4