Given [(x-5)/(x-3)]^2 . Find any stationary points and any points of inflection. Also find any horizontal and
vertical asymptotes.
Stationary points are points at which f'(x)=0 AND f"(x)=0.
Vertical asymptote is where the denominator becomes zero.
Horizontal asymptote is the limit when x->∞ or x->-∞, if the limit is defined.
could you please solve the problem completely .. i am still confused.
Thanks.
Denominator becomes zero when x=3, therefore vertical asymptote is a x=3.
The limit of f(x)=[(x-5)/(x-3)]^2 as x->±∞ is 1, so the horizontal asymptote is at y=1.
Since f'(5)=0 and f"(6)=0, there are no stationary points, just a minimum at x=5.
There is a point of inflection at x=6 (where f"(6)=0).
You will need to take the time to calculate f'(x) and f"(x) from f(x), and hence sketch the graph of f(x).
A sketch of the graph can be seen here:
http://img718.imageshack.us/img718/4296/1292787276.png
To find the stationary points and points of inflection, as well as the horizontal and vertical asymptotes, we need to analyze the function represented by the given expression \(\left(\frac{x-5}{x-3}\right)^2\).
First, let's determine the stationary points. A stationary point occurs when the derivative of the function is equal to zero.
1. Find the derivative of the function:
To do this, we need to apply the chain rule. Let's call the given expression \(y\):
\[y = \left(\frac{x-5}{x-3}\right)^2\]
To avoid dealing with fractions, let's simplify this expression:
\[y = \left(\frac{(x-5)^2} {(x-3)^2}\right)\]
Now we can differentiate y with respect to x:
\[\frac{dy}{dx} = \frac{d}{dx}\left(\frac{(x-5)^2} {(x-3)^2}\right)\]
Using the quotient rule, the derivative can be calculated as follows:
\[\frac{dy}{dx} = \frac{(x-3)^2\cdot2(x-5) - (x-5)^2\cdot2(x-3)}{(x-3)^4}\]
2. Find the stationary points:
To find the stationary points, set the derivative equal to zero and solve for x:
\[\frac{dy}{dx} = 0\]
\[\frac{(x-3)^2\cdot2(x-5) - (x-5)^2\cdot2(x-3)}{(x-3)^4} = 0\]
\[(x-3)^2\cdot2(x-5) - (x-5)^2\cdot2(x-3) = 0\]
Simplifying this equation will lead to the solution(s) for the stationary point(s) of the function.
Now let's find the points of inflection:
1. Find the second derivative:
To find the points of inflection, we need to examine the concavity of the function by finding the second derivative.
\[\frac{d^2y}{dx^2} = \frac{d}{dx} \left(\frac{dy}{dx}\right)\]
To simplify this expression, we can rewrite the first derivative obtained earlier.
\[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left(\frac{(x-3)^2 \cdot 2(x-5) - (x-5)^2 \cdot 2(x-3)}{(x-3)^4}\right)\]
Simplify the expression to get the second derivative.
2. Find the points of inflection:
Set the second derivative equal to zero and solve for x to find the points of inflection.
Finally, let's find the horizontal and vertical asymptotes:
1. Horizontal asymptotes:
To find the horizontal asymptote(s), we need to analyze the behavior of the function as x approaches positive or negative infinity. By examining the highest power of x in both the numerator and denominator, we can determine the horizontal asymptote(s).
2. Vertical asymptotes:
Vertical asymptotes represent the values of x for which the function is undefined. In this case, the denominator should be zero. Solve the equation \(x-3=0\) to find the vertical asymptote(s) for the function.
By following these steps, you should be able to find the stationary points, points of inflection, as well as the horizontal and vertical asymptotes of the given expression.