Suppose that f(x)=((x-4)/(x^2))

Question

c.) Find all critical values of f. Critical value(s)=

d.) Use interval notation to indicate where f(x) is increasing or decreasing.

Increasing: Decreasing:

e.) Find the x-coordinates of all local maxima and minima of f.

Local maxima at x=

Answer 1

The best way to answer this question is to graph the function.

You will immediately see:,

>f blows up negatively at x = 0

>f is asymptotic to y = 0 from below as x goes to - infinity

>x = 4 is a zero where f crosses from negative to positive and then is asymptotic to y=0 from above as x goes to positive infinity

f''(x)= (x³-2x²+8x)/x⁴

and as long as x is not equal to 0 this is (x²-2x+8)/x³

The numerator has no real zeros.

However, there appears to be a local maximum near x=8, but I am unable to find it analytically; I will think about the problem and get back to you if I figure it out.