Find the intercept, relative extrema, point of inflection, and asymptotes of the function(if they exist)of:
y = (x^3)/(x^2 - 4)
To find the intercepts of the function y = (x^3)/(x^2 - 4), we need to set y equal to zero and solve for x.
1. x-intercept:
Setting y = 0, we have:
0 = (x^3)/(x^2 - 4)
Since the numerator (x^3) is zero when x = 0, we need to check for any restrictions or exclusions on the denominator (x^2 - 4).
x^2 - 4 ≠ 0
x^2 ≠ 4
x ≠ ±2
So, there are no x-intercepts for this function.
2. y-intercept:
To find the y-intercept, we set x = 0 and solve for y:
y = (0^3)/(0^2 - 4)
y = 0/(-4)
y = 0
Therefore, the y-intercept is (0, 0).
Next, let's find the relative extrema of the function.
To find the relative extrema, we need to find the critical points. These occur when the derivative of the function is equal to zero or does not exist.
First, we find the first derivative of y with respect to x:
dy/dx = [(3x^2)(x^2 - 4) - (x^3)(2x)] / (x^2 - 4)^2
Now, we equate the derivative to zero and solve for x:
[(3x^2)(x^2 - 4) - (x^3)(2x)] / (x^2 - 4)^2 = 0
Simplifying the equation gives us:
[3x^4 - 12x^2 - 2x^4] / (x^2 - 4)^2 = 0
x^4 - 12x^2 = 0
x^2(x^2 - 12) = 0
From this equation, we find two critical points:
x^2 = 0 => x = 0
x^2 - 12 = 0 => x = ±√12
x = ±2√3
Now, let's determine the corresponding y-values at these critical points.
For x = 0:
y = (0^3)/(0^2 - 4)
y = 0
For x = 2√3:
y = ((2√3)^3)/((2√3)^2 - 4)
y = (8√3)/(12 - 4)
y = (8√3)/8
y = √3
For x = -2√3:
y = ((-2√3)^3)/((-2√3)^2 - 4)
y = (-8√3)/(12 - 4)
y = (-8√3)/8
y = -√3
Therefore, we have two relative extrema:
Relative maximum: (√3, √3)
Relative minimum: (-√3, -√3)
Next, let's find the point of inflection.
The point of inflection occurs where the second derivative of the function is equal to zero or does not exist.
First, we find the second derivative of y with respect to x:
d^2y/dx^2 = [6x(x^2 - 4)(x^2 - 6) - (x^3)(2)(2x)] / (x^2 - 4)^3
Now, we need to determine where the second derivative is equal to zero or undefined.
[6x(x^2 - 4)(x^2 - 6) - (x^3)(2)(2x)] / (x^2 - 4)^3 = 0
Simplifying and factoring, we have:
6x(x^2 - 4)(x^2 - 6) - 4x^4 = 0
6x(x^2 - 4)(x^2 - 6) = 4x^4
Since 4x^4 is always positive or zero, for this equation to hold true, the left side must also always be positive or zero.
Therefore, there are no points of inflection for this function.
Lastly, let's find any asymptotes.
1. Vertical asymptotes:
The vertical asymptotes occur where the denominator of the function becomes zero.
So, let's find the values of x where x^2 - 4 = 0:
x^2 - 4 = 0
(x - 2)(x + 2) = 0
x = ±2
Hence, there are two vertical asymptotes at x = 2 and x = -2.
2. Horizontal asymptote:
To determine the horizontal asymptote, we analyze the behavior of the function as x approaches positive or negative infinity.
As x goes to positive or negative infinity, the term (x^3)/(x^2 - 4) dominates the function. As a result, the function approaches the value of (x^3)/(x^2) = x.
Therefore, the horizontal asymptote is y = x.
In summary, for the given function y = (x^3)/(x^2 - 4):
- There are no x-intercepts.
- The y-intercept is (0, 0).
- The relative extrema are the relative maximum at (√3, √3) and the relative minimum at (-√3, -√3).
- There are no points of inflection.
- Vertical asymptotes are x = 2 and x = -2.
- The horizontal asymptote is y = x.