Consider the following.
f(x) = 8x(square root of (x − x^2)) Use a graph to find the absolute maximum and minimum values of the function to two decimal places.
only valid for 90 ≤ x ≤ 1
http://www.wolframalpha.com/input/?i=y+%3D+%3D+8x%28√%28x+−+x%5E2%29%29+%2C+0+%3C+x+%3C+1
only valid for 0 ≤ x ≤ 1
dy/dx = 8x(1/2)(x-x^2)^(-1/2) (1 - 2x) + 8(x-x^2)^(1/2)
= 4(x-x^2)^(-1/2) [ x(1-2x) + 2(x-x^2) ]
= (4/√(x-x^2) ( x - 2x^2 + 2x - 2x^2)
= (4/√(x-x^2) (3x - 4x^2)
= 0 for a max/min
3x - 4x^2 = 0
x(3 - 4x) = 0
x = 0 ----> looks like it will yield a min
x = 3/4 --->looks like it will yield a max
f(0) = 0
f(3/4)
=8(3/4)√(3/4 - 9/16)
= 6√(3/16)
= 6√3/4 = 3√3/2 or 2.598
min is 0
max is aprr 2.60
http://www.wolframalpha.com/input/?i=maximum+of+8x%28√%28x+−+x%5E2%29%29+
hover your cursor over the red point to show its coordinates
To find the absolute maximum and minimum values of the given function f(x), we can use a graphing calculator or a graphing software.
Here's an explanation of how to use a graph to find the absolute maximum and minimum values:
1. Plot the graph of the function: Start by plotting the graph of the function f(x) = 8x√(x − x^2). You can use a graphing calculator, a graphing software, or an online graphing tool to plot the graph.
2. Analyze the graph: Once you have the graph plotted, examine it to identify the highest and lowest points on the graph. These points will correspond to the absolute maximum and minimum values of the function.
3. Determine the x-values: To find the x-values corresponding to the highest and lowest points, locate the x-coordinates of those points on the graph.
4. Plug the x-values into the function: Once you have the x-values, substitute them back into the original function f(x) = 8x√(x − x^2) to find the corresponding y-values.
5. Round to two decimal places: Finally, round the obtained y-values to two decimal places to get the absolute maximum and minimum values of the function.
Note: Since you didn't specify the interval or range for the x-values, the absolute maximum and minimum values you find will be applicable within the given domain or the overall domain of the function.
By following these steps, you can find the absolute maximum and minimum values of the function f(x) = 8x√(x − x^2) using a graph.