y = 4x -4 / x^2
vertical asymptotes: ____________________
horizontal asymptotes: ____________________
increasing intervals: ____________________
decreasing intervals: ____________________
relative maxima: ____________________
relative minima: ____________________
concave up intervals: ____________________
concave down intervals: ____________________
inflection points: _________________
To find the vertical asymptotes, we need to determine the values of x for which the expression y = 4x - 4 / x^2 becomes undefined. In this case, the expression becomes undefined when the denominator, x^2, equals 0. Therefore, we set x^2 = 0 and solve for x.
x^2 = 0
x = 0
So, the vertical asymptote is x = 0.
To determine the horizontal asymptote, we need to analyze the behavior of the expression as x approaches positive infinity and negative infinity.
As x approaches positive infinity or negative infinity, the term 4x becomes dominant compared to the term -4 / x^2. Therefore, the expression can be approximated as y ≈ 4x / x^2.
Simplifying further, we get y ≈ 4 / x.
As x approaches positive infinity, the expression 4 / x approaches 0. Therefore, the horizontal asymptote is y = 0.
To find the increasing intervals, we need to determine the intervals on the x-axis where the function is increasing. This happens when the derivative of the function is positive.
Taking the derivative of the function y = 4x - 4 / x^2, we get:
y' = (4 * x^2 - (-4) * 2x) / (x^2)^2
y' = (4x^2 + 8x) / x^4
y' = (4x(x + 2)) / x^4
Setting the derivative equal to 0 and solving for x:
4x(x + 2) / x^4 = 0
Since the numerator is never equal to 0, the derivative is never equal to 0. Therefore, the function does not have any relative maxima or minima.
To determine the concavity of the function, we need to analyze the second derivative. Taking the derivative of the derivative, we get:
y'' = (4(x + 2) * x^4 - 4x^2(4x)) / x^8
y'' = (4x^5 + 8x^4 - 16x^3) / x^8
y'' = (4x^3(x^2 + 2x - 4)) / x^8
Setting the second derivative equal to 0 and solving for x:
4x^3(x^2 + 2x - 4) / x^8 = 0
Since the numerator is never equal to 0, the second derivative is never equal to 0. Therefore, the function does not have any inflection points.
To summarize our findings:
Vertical asymptotes: x = 0
Horizontal asymptote: y = 0
Increasing intervals: None
Decreasing intervals: None
Relative maxima: None
Relative minima: None
Concave up intervals: None
Concave down intervals: None
Inflection points: None