Find the intercept, relative extrema, point of inflection, and asymptotes of the function(if they exist)of:
y = (x^2 + 1)/(x^2 - 9)
The intercept is the y value when x = 0. That's pretty easy.
The relative extrema occur where dy/dx = 0
The inflection point is where d^2y/dx^2 = 0
The asymptotes are where the denominator of y(x) is zero: x = 3 and -3.
Now you have some differentiating to do.
y=0^2+1/0^2+9 y = 1/9 y intecept
To find the intercepts of the function y = (x^2 + 1)/(x^2 - 9), we set y to zero and solve for x.
Setting y = 0, we have:
(x^2 + 1)/(x^2 - 9) = 0
To make the numerator zero, x^2 + 1 must be zero. However, x^2 + 1 has no real solutions because no real number squared can be negative. Therefore, the function has no x-intercept.
To find the y-intercept, we set x to zero and solve for y.
Setting x = 0, we have:
y = (0^2 + 1)/(0^2 - 9) = 1/(-9) = -1/9
Therefore, the y-intercept is (0, -1/9).
To find the relative extrema, we take the derivative of the function and find the critical points.
First, let's simplify the function:
y = (x^2 + 1)/(x^2 - 9) = (1 + x^2)/(x^2 - 9)
To find the derivative, we can use the quotient rule:
dy/dx = [(1 + x^2)(2x) - (x^2 - 9)(2x)] / (x^2 - 9)^2
To find the critical points, we set the numerator equal to zero:
(1 + x^2)(2x) - (x^2 - 9)(2x) = 0
Simplifying, we have:
2x + 2x^3 - 2x^3 + 18x = 0
4x^2 + 18x = 0
2x(2x + 9) = 0
The critical points are x = 0 and x = -9/2.
To determine if these critical points are relative extrema, we can use the second derivative test. Taking the second derivative of the function, we have:
d²y/dx² = [(2x + 9)(x^2 - 9)^2 - (4x)(2x)(x^2 - 9)] / (x^2 - 9)^4
Evaluating the second derivative at x = 0 and x = -9/2, we can determine the nature of the critical points:
For x = 0:
d²y/dx² = (9)(-9)^2 / (-9)^4 = 1/9
Since the second derivative is positive, x = 0 is a relative minimum point.
For x = -9/2:
d²y/dx² = (-9/2 + 9)(-9/2 - 9)^2 - (4(-9/2))(2(-9/2))((-9/2)^2 - 9)] / ((-9/2)^2 - 9)^4
Evaluating the above expression, the second derivative is negative, indicating that x = -9/2 is a relative maximum point.
Therefore, the relative extrema are:
Relative minimum: (0, -1/9)
Relative maximum: (-9/2, 5/9)
To find the points of inflection, we set the second derivative equal to zero and solve for x:
(2x + 9)(x^2 - 9)^2 - (4x)(2x)(x^2 - 9) = 0
Simplifying this equation may not lead to a simple solution. However, it can be solved numerically using a graphing calculator or software.
Lastly, let's explore asymptotes. As x approaches positive or negative infinity, the function y = (x^2 + 1)/(x^2 - 9) approaches the horizontal asymptote y = 1. This can be determined by analyzing the leading terms of the function.
Therefore, the function has a y-intercept at (-1/9, 0), a relative minimum at (0, -1/9), a relative maximum at (-9/2, 5/9), and a horizontal asymptote at y = 1. The points of inflection can be found numerically.