How do I solve this Fundamental Theorem of Calculus problem?

Question

Consider the function f(x)=((x²−1)/x

and define the function F(x)=∫from1tox f(t)dt

with domain (0,∞)

a) What are critical points of F?

b) Find the intervals of increasing and decreasing for F.

c) Find the global maximum and minimum of F

on the interval [1/2,3].

Answer 1

a) Fundamental Thm of Calculus tells you that F'(x) = f(x). Therefore set f(x) = 0 to find critical points of F: (x²-1)/x is 0 when x = 1.

b) f is negative on (0, 1), so F is decreasing there

f is positive on (1, ∞) so F is increasing on (1, ∞)

c) For absolute min and max, evaluate F at critical point (there is only one of those) and endpoints of domain. Compare this finite set of function values to find the biggest and the smallest)

F(1/2) = t²/2 - ln(t) evaluated at 1 (lower limit) and 1/2 (upper limit)

= (1/8 - ln(1/2)) - (1/2 - ln(1)) = -3/8 - (-ln(2)) ≈ 0.318

F(1) = 0 (upper and lower limits of integration are the same, so result is 0)

F(3) = 3²/2 - ln(3) - (1/2 - ln(1)) = 9/2 - ln(3) - 1/2 = 4 - ln(3) ≈ 2.901

Therefore 0 = F(1) is the absolute min

2.901 ≈ F(3) is the absolute max