Question

The derivative of a function f is f'(x) = x^2 - 8/x: Then at x = 2, f has
A. an infection point B. a relative maximum C. a vertical tangent
D. a discontinuity E. a relative minimum

how do you test for all of these?

Answer 1

f'(2) = 4 - 8/2 = 0
f"(x) = 2x+8/x^2
f"(2) = 4 + 8/4 = 6

so, f'(2)=0 and f"(2)>0, means there is a local minimum there.

Recall that an inflection point is where f" = 0 (an infection point is just sick!)

a discontinuity is where f is undefined.

a vertical tangent is where f is defined but f' is undefined

Answer 2

thanks. so how would u see if f is undefined? how do you get f from f'?

Answer 3

when dealing with rational functions, the value is undefined when the denominator is zero.

So, f' and f" are undefined at x=0

f is the anti-derivative of f'. So,
f(x) = 1/3 x^3 - 8ln(x) + c

If you take the derivative of that, you'll get f' as given. Note that the value of c does not matter, since the derivative of a constant is zero.