# Thread: Critical and Inflection Points of f(x)

1. ## Critical and Inflection Points of f(x)

f(x) = x + 1/x
f(x) = x + x^-1

f'(x) = 1 - x^-2
f'(x) = 1 -1/x^2

f"(x) = 2x^-3
f"(x) = 2/x^3

To find critical point value of x, I set f'(x) = 0 and solve for x
To find inflection point value of x, I set f"(x) = 0 and solve for x.
Unfortunately, this problem doesn't make solving for x very easy.

2. Originally Posted by confusedagain
f(x) = x + 1/x
f(x) = x + x^-1

f'(x) = 1 - x^-2
f'(x) = 1 -1/x^2

f"(x) = 2x^-3
f"(x) = 2/x^3

To find critical point value of x, I set f'(x) = 0 and solve for x
To find inflection point value of x, I set f"(x) = 0 and solve for x.
Unfortunately, this problem doesn't make solving for x very easy.
Let us see.

f'(x) = 1 -1/(x^2)
Set that to zero,
0 = 1 -1/(x^2)
Clear the fractions, multiply both sides by x^2
0 = x^2 -1
0 = (x+1)(x-1)
So,
x = 1 or (-1) -----------------**

Or,
0 = x^2 -1
x^2 = 1
x = +,-sqrt(1)
x = 1 or (-1) --------------same.

---------------------------------------------
f''(x) = 2/(x^3)
Set that to 0,
0 = 2/(x^3)
Clear the fractions, multiply both sides by x^3,
0 = 2 -------------yeah, right.
Meaning, there is no inflection point.
You have and know how to use a graphing calculator? See the graph of f(x) = 1 +1/x. Maybe the graph really does not have an inflection point.