Find the absolute maximum value and the absolute minimum value, if any, of the function.
g(x) = -x(^2) + 2 x + 9
g'(x) = -2x + 2
This is zero when x = 1.
The second derivative, -2, is negative there, so g(1) is a maximum. The value there is 10.
There is no absolute minimum. The function is an upside-down parabola that goes to minus infinity at large and small x.
In google type:
functions graphs online
When you see list of results click on:
rechneronline.de/function-graphs/
When page be open in blue recatacangle type:
-x^2 + 2 x + 9
In Display properties set:
Range y-axis from -7 to 13
Then click option Draw
You will see graph of your function.
To find the absolute maximum and minimum values of the function g(x) = -x^2 + 2x + 9, we can follow these steps:
Step 1: Take the derivative of the function g(x) with respect to x.
g'(x) = -2x + 2
Step 2: Set the derivative equal to zero and solve for x to find critical points.
-2x + 2 = 0
2x = 2
x = 1
Step 3: Determine the nature of the critical point(s) by evaluating the second derivative.
Taking the second derivative of g(x), we get:
g''(x) = -2
Since g''(x) is a constant (-2), we can conclude that the critical point at x = 1 is a local maximum.
Step 4: Evaluate the function at the critical point(s) and the endpoints of the interval, if given.
Let's also calculate the values of g(x) at x = 1 and at the endpoints of the interval.
g(1) = -(1^2) + 2(1) + 9
g(1) = -1 + 2 + 9
g(1) = 10
Step 5: Determine if there are any endpoints to consider.
Since there are no specified endpoints mentioned, we will assume that the domain of the function is the set of all real numbers (-∞, +∞).
Step 6: Determine the absolute maximum and minimum values.
Since we are considering the entire domain of real numbers, there are no absolute maximum or minimum values for the function g(x) = -x^2 + 2x + 9.
Therefore, the function g(x) = -x^2 + 2x + 9 does not have any absolute maximum or minimum values.
To find the absolute maximum and minimum values of a function, we need to analyze the critical points and endpoints of the function. Here's how we can do it step by step:
Step 1: Take the derivative of the function to find the critical points.
The derivative of g(x) = -x² + 2x + 9 can be found using the power rule of differentiation:
g'(x) = -2x + 2
Step 2: Set the derivative equal to zero and solve for x to find the critical points.
-2x + 2 = 0
Solving this equation, we get:
x = 1
Step 3: Determine the value of g(x) at the critical points.
To find the value of g(x) at x = 1, we substitute the value into the original function:
g(1) = -(1²) + 2(1) + 9
g(1) = -1 + 2 + 9
g(1) = 10
Step 4: Identify the endpoints of the function.
Since there are no specific constraints mentioned for the domain of the function g(x), we assume it extends to positive and negative infinity. Therefore, there are no endpoints to consider.
Step 5: Analyze the values of g(x) at the critical points and endpoints to determine the absolute maximum and minimum values.
From step 3, we have the value of g(x) at x = 1, which is g(1) = 10.
Since there are no endpoints to consider, the only value we need to analyze is g(1).
Hence, the absolute maximum value of g(x) is 10, and there are no absolute minimum values.
Note: It's important to note that in cases where the domain of the function is restricted, we need to examine the endpoints as well to determine the absolute maximum and minimum values.