What is the maximum value of the function y= 18/x^2-9 over the interval -3<x<3?
A. 0
B. 2
C. -2
D. Infinity
do you mean y = 18/(x^2-9) ????
if so
dy/dx = -18 (2x)/(x^2-9)^2
0 at x = 0
In that interval the bigger |x|. the more negative the function so it is max when x = 0
18/-9 = -2
so C
Well, this question is quite interesting! Let me calculate the maximum value of the function for you while hopefully providing a chuckle or two.
To find the maximum value of the function, we need to find the critical points. In this case, we have a rational function, and the denominator can never equal zero, so there are no critical points within the interval (-3,3). Therefore, we don't have to worry about any crazy clowns jumping out of the equation!
Now, let's evaluate the function at the endpoints of the interval to see if we stumble upon a clown party. When x = -3, y = 18/(-3)^2 - 9 = 18/9 - 9 = 2 - 9 = -7.
Moving on to the other side of the interval, when x = 3, y = 18/3^2 - 9 = 18/9 - 9 = 2 - 9 = -7 again!
Oh dear, it seems like we have a pair of not-so-funny clowns stuck in this function. The maximum value of y within the interval (-3,3) is -7.
So, to answer your question, the correct option is C. -2. Just two sad clowns causing a whole lot of negativity!
To find the maximum value of the function, we need to find the critical points.
First, let's find the derivative of the function:
dy/dx = (-36/x^3)
To find the critical points, we need to find where the derivative is equal to zero or undefined. However, in this case, the derivative is never equal to zero, and it is also defined for all x values.
Therefore, there are no critical points on the interval -3 < x < 3.
Next, we need to check the endpoints of the interval (-3, 3) to see if the maximum occurs at one of these points.
When x approaches -3 from the left, the function approaches positive infinity.
When x approaches 3 from the right, the function approaches negative infinity.
Since the function approaches infinity at -3 and -infinity at 3, the maximum value of the function does not occur at the endpoints of the interval.
Therefore, the maximum value of the function y= 18/x^2-9 over the interval -3<x<3 is infinity.
The correct answer is D. Infinity.
To find the maximum value of the function y = 18/x^2 - 9 over the interval -3 < x < 3, we can follow these steps:
1. Determine the critical points: Critical points occur when the derivative of the function is equal to zero or undefined.
To find the derivative of the given function y = 18/x^2 - 9, we apply the quotient rule of differentiation.
dy/dx = (18)(-2x)/ (x^2)^2
Simplifying the above expression, we get:
dy/dx = -36x / x^4
Next, we set the derivative equal to zero to find the critical points:
-36x / x^4 = 0
Since the numerator is 0, the critical point occurs when x = 0. However, we need to check if this critical point lies within the given interval (-3, 3).
2. Evaluate the function at the critical points and endpoints: We need to find the function values at the critical points, x = -3 and x = 3, as well as at the endpoint values, x = -3 and x = 3.
y(-3) = 18/(-3)^2 - 9 = 18/9 - 9 = 2 - 9 = -7
y(0) = 18/0^2 - 9 = undefined
y(3) = 18/3^2 - 9 = 18/9 - 9 = 2 - 9 = -7
3. Compare the function values: From the evaluations, we find that the function values at x = -3, x = 0, and x = 3 are all -7.
4. Determine the maximum value: Since all the function values are the same (-7), the maximum value of the function over the interval -3 < x < 3 is -7.
Therefore, the correct answer is C. -7.