Given [(x-5)/(x-3)]^2 . Find any stationary points and any points of inflection. Also find any horizontal and
vertical asymptotes.
See:
http://www.jiskha.com/display.cgi?id=1292787276
To find the stationary points of a function, we need to find the values of x where the derivative of the function is equal to 0. Similarly, to find the points of inflection, we need to find the values of x where the second derivative of the function is equal to 0. Let's start by finding the derivatives.
Given function: [(x-5)/(x-3)]^2
Step 1: Expand the expression.
[(x-5)/(x-3)]^2 = (x-5)^2/(x-3)^2
Step 2: Find the first derivative.
Let f(x) = (x-5)^2/(x-3)^2
To find f'(x), we can use the quotient rule:
f'(x) = [2(x-5)(x-3)^2 - 2(x-5)^2(x-3)]/(x-3)^4
Simplifying further, we can obtain:
f'(x) = 4(x-5)(x-3)/[(x-3)^4]
Step 3: Find the second derivative.
To find the second derivative, we differentiate f'(x) using the quotient rule again.
Let g(x) = 4(x-5)(x-3)/[(x-3)^4]
Using the quotient rule, we get:
g'(x) = [(4[(x-5)(x-3)^4]' - (x-5)(x-3)^4[4(x-3)]) / [(x-3)^8]
Simplifying further, we have:
g'(x) = [12(x-3)^4 - 4(x-5)(x-3)^3] / [(x-3)^8]
Now, to find the stationary points, we need to solve f'(x) = 0.
4(x-5)(x-3) / [(x-3)^4] = 0
From this equation, we can see that the numerator must be equal to zero:
4(x-5)(x-3) = 0
This gives us two possible solutions:
x - 5 = 0 -> x = 5
x - 3 = 0 -> x = 3
So, there are two stationary points: x = 5 and x = 3.
To find the points of inflection, we need to solve g'(x) = 0.
[12(x-3)^4 - 4(x-5)(x-3)^3] / [(x-3)^8] = 0
From this equation, we can see that the numerator must be equal to zero:
12(x-3)^4 - 4(x-5)(x-3)^3 = 0
Simplifying the equation further, we get:
12(x-3)^4 = 4(x-5)(x-3)^3
Dividing both sides by (x-3)^3, we have:
12(x-3) = 4(x-5)
Simplifying this equation further, we get:
12x - 36 = 4x - 20
Solving for x, we find:
8x = 16
x = 2
So, there is one point of inflection: x = 2.
Now, let's find the horizontal and vertical asymptotes.
Horizontal Asymptotes:
To find horizontal asymptotes, we need to determine the behavior of the function as x approaches positive or negative infinity.
As x approaches positive or negative infinity, the function can be simplified to:
f(x) = (x-5)^2/(x-3)^2
As x approaches positive infinity, the terms with higher powers (x^2 terms) become dominant. Therefore, the function approaches the value of 1.
As x approaches negative infinity, the negative x^2 terms become dominant, and the function still approaches the value of 1.
So, the horizontal asymptote is y = 1.
Vertical Asymptotes:
To find vertical asymptotes, we need to determine the values of x for which the denominator of the expression becomes zero.
In this case, the denominator is (x-3)^2. Setting it equal to zero, we have:
(x-3)^2 = 0
This gives us a vertical asymptote at x = 3.
In summary:
- The stationary points are x = 5 and x = 3.
- The point of inflection is x = 2.
- The horizontal asymptote is y = 1.
- The vertical asymptote is x = 3.