Let h be a function defined for all x≠0 such that h(4)=-3 and the derivative h is given by

Question

h'(x)=(x^2-2)/(x) for all x≠0
a). Find all values of x for which the graph of h has a horizontal tangent, and determine whether h has a local maximum, a local minimum, or neither at each of these values. Justify your answers
b) On what intervals, if any, is the graph of h concave up? Justify your answers

Answer 1

clearly the tangent is horizontal when

x^2-2 = 0

h"(x) = (x^2+2)/x^2

And you know that f(x) is concave up when f"(x) > 0

See that graph at

http://www.wolframalpha.com/input/?i=integral+x+-+2%2Fx