Identify the open intervals on which the function is increasing or decreasing

Question

(a) f(x)= x^4-2x^2

(b) f(x)= x/x+1

Answer 1

a) f'(x)=4x³-4x=4x(x-1)(x+1).

Three points of interest: x=-1, x=0 and x=1. The curve resembles a W shape, with minima at x=-1 and x=1.

f''(x)=12x²-4=4(3x²-1), f''(-1) = f''(1) > 0 confirming minima. f''(0)<0 confirming maximum at x=0.

x<-1 concave down, -1<x<0 concave up, 0<x<1 concave down, x>1 concave up.

b) f(x)=x/(x+1) is vertically asymptotic at x=-1. When x is large and negative f(x)→1, when x is large and positive f(x)→1, so f(x)=1 is a horizontal asymptote.

Between x=-∞ and -1 the curve is upward concave (increasing), first of all gradually then suddenly very steeply, and between x=-1 and ∞ the curve is downward concave (decreasing), first of all declining steeply then more gradually. It is undefined at x=-1 hence the asymptote.