G(x)= x√9-x
find x-intercept, y-intercept
Vertical, horizontal asymptotes
Extremes
Points of inflection
i need to find all of these.
Is the 9-x supposed to be all under the square root sign?
Set G(x) = 0 to get the x-intercept. Set x = 0 to get the y intercept.
Set dG/dx = 0 to get the extremes and inflection points. If d^2G/dx^2 = 0 when dG/dx = 0, it is an inflection point.
We'll be glad to critique your work.
First of all, it is spelled calculus.
Next, show your work so that we know where you need help.
Okay?
First of all, it is spelled calculus.
Next, show your work so that we know where you need help.
Okay?
To find the x-intercept of a function, we set the value of the function equal to zero and solve for x. Similarly, to find the y-intercept, we can set x equal to zero and evaluate the function.
1. X-intercept: Set G(x) = 0 and solve for x:
x√(9-x) = 0
Since multiplication is zero if and only if one (or both) of the factors is zero, we have two cases to consider:
Case 1: x = 0
Case 2: √(9-x) = 0
9 - x = 0
x = 9
Therefore, the x-intercepts are x = 0 and x = 9.
2. Y-intercept: Set x = 0 in G(x):
G(0) = 0√(9-0) = 0√9 = 0
Therefore, the y-intercept is y = 0.
To find the vertical asymptotes, we look for values of x that make the function undefined. In this case, the function G(x) is defined for all real numbers.
To find the horizontal asymptote, we examine the behavior of the function as x approaches negative infinity and positive infinity.
3. Horizontal asymptote: As x approaches positive or negative infinity, the term involving the square root becomes negligible compared to the term involving x. Therefore, the function approaches the horizontal line y = 0. Hence, the horizontal asymptote is y = 0.
To find the extremes of the function, we need to find the critical points by taking the derivative of G(x) and setting it equal to zero.
4. Critical points (Extremes):
G(x) = x√(9-x)
G'(x) = √(9-x) - x/(2√(9-x))
Set G'(x) = 0 and solve for x:
√(9-x) - x/(2√(9-x)) = 0
To simplify this equation, let's multiply through by 2√(9-x):
2(9-x) - x = 0
18 - 2x - x = 0
18 - 3x = 0
-3x = -18
x = 6
We can now substitute x = 6 back into the original function to find the y-coordinate (extreme value):
G(6) = 6√(9-6) = 6√3
Therefore, the extreme point is (6, 6√3).
To find the points of inflection, we take the second derivative of G(x) and set it equal to zero.
5. Points of inflection:
G(x) = x√(9-x)
G''(x) = -1/(2√(9-x))
Set G''(x) = 0 and solve for x:
-1/(2√(9-x)) = 0
There are no values of x that satisfy this equation. Therefore, there are no points of inflection in this function.
To summarize:
- X-intercepts: x = 0 and x = 9
- Y-intercept: y = 0
- Vertical asymptotes: None
- Horizontal asymptote: y = 0
- Extremes: (6, 6√3)
- Points of inflection: None