Finding critical numbers, determine increase/decrease, concave up/down calculus 1

Question

Consider the function f(x)=4x+7x^-1. For this function there are four important intervals: (−∞,A], [A,B),(B,C], and [C,∞) where , and are the critical numbers and the function is not defined at .

1.Find A, B, and C

2.At each important interval tell if it is increasing or decreasing.

3.~Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x) is concave up or concave down.

(-infinity, B)

(B, infinity) Please show detailed work for all solutions.

Answer 1

f(x) = 4x+7/x

f '(x) = 4-7/x²

Solving f '(x) =0 we get x= -1.32, 1.32 and x=0 the function is undefined.

These are the three critical numbers.

A= -1.32, B =0 , C = 1.32

Checking signs of f'(x) in the three intervals we find f(x) is inc in (−∞,A), dec in (A,B) and (B,C) , inc again (C,∞)

f ''(x) = 14/x³ which is never zero.

but sign of f''(x)<0 from (−∞,B), hence concave down.

f"(X) >0 in (B, ∞) hence concave up