If someone could help me on this, I'd appreciate it.
f(x)= 2x/(x^2+1)
Vertical: none
Horizontal: none
Minimum: x=-1
Maximum: x=1
Inflection Points: it seems like there are one or two, but i don't know how to factor f''(x)= 4x^3 + x^2 - 12x + 1 :/
Wolfram confirms the following:
http://www.wolframalpha.com/input/?i=2x%2F%28x%5E2%2B1%29+
No VA
No HA
max =1
min = 1
to get 2nd derivative
Wolfram said:
4x(x^2 - 3)/(x^2 + 1)^3
http://www.wolframalpha.com/input/?i=-2%28x%5E2+-1%29%2F%28x%5E2%2B1%29%5E2
So check your work for the second derivative
for points of inflection:
4x(x^2 - 3)/(x^2 + 1)^3 = 0
4x(x^2 - 3) = 0
so x = 0 or x^2 = 3
x = 0 or x + √3 or x = -√3
if x = 0 , y = 0 --> (0,0)
if x = √3 , y = 2√3/5 --> (2√3/5)
if x = -√3 -----------> (-2√3/5)
this is confirmed by looking of the graph in the first Wolfram link
Thank you so much! I see what I should have done now. :P
To find the inflection points of a function, we need to find the values of x where the second derivative changes sign. In this case, we need to find the values of x for which f''(x) = 4x^3 + x^2 - 12x + 1 changes sign.
To find the inflection points, we can follow these steps:
Step 1: Find f''(x)
Differentiate f(x) with respect to x, using the power rule and quotient rule:
f'(x) = (2(x^2+1) - 2x(2x))/(x^2+1)^2
Simplify f'(x):
f'(x) = (2x^2 + 2 - 4x^2)/(x^2+1)^2
f'(x) = (2 - 2x^2)/(x^2+1)^2
Step 2: Find f''(x)
Differentiate f'(x) with respect to x, using the quotient rule:
f''(x) = ((2x^2 + 2 - 4x^2)(2(x^2+1)^2) - (2 - 2x^2)(2(x^2+1))(2x))/((x^2+1)^2)^2
Simplify f''(x):
f''(x) = (2x^2 + 2 - 4x^2)(2(x^2+1)^2 - (2 - 2x^2)(2(x^2+1))(2x))/(x^2+1)^4
f''(x) = (2x^2 + 2 - 4x^2)(2x^4 + 4x^2 + 2) - (2 - 2x^2)(4x^3 + 8x^2 + 4x))/(x^2+1)^4
f''(x) = (2x^6 + 4x^4 + 2x^2 + 4x^2 + 8x^2 + 4 - 8x^4 - 16x^2 - 8x - 8x^2 + 8x^4 + 8x^2 + 4x^3 + 8x^2 + 4x)/(x^2+1)^4
f''(x) = (2x^6 - 8x^4 + 8x^4 + 4x^3)/(x^2+1)^4
f''(x) = (2x^6 + 4x^3)/(x^2+1)^4
f''(x) = 2x^3(x^3 + 2)/(x^2+1)^4
Step 3: Solve for f''(x) = 0
Set f''(x) equal to 0 and solve for x:
2x^3(x^3 + 2)/(x^2+1)^4 = 0
Since the numerator is never equal to 0, the only solution is x = 0.
Therefore, the only inflection point of the function f(x) = 2x/(x^2+1) is at x = 0.