Find the absolute maximum value and the absolute minimum value, if any, of the function.
g(x)=1/8x^2-4�ã(x)on [0,49]
maximum ?
minimum ?
To find the maximum and minimum values of the function g(x) = 1/8x^2 - 4�‾(x) on the interval [0, 49], we need to follow these steps:
Step 1: Find the critical points of the function by setting g'(x) = 0.
Step 2: Determine the endpoints of the interval [0, 49].
Step 3: Evaluate the function at the critical points and the endpoints.
Step 4: Compare the values obtained to find the absolute maximum and minimum.
Let's start with step 1.
Step 1: Find the critical points by setting g'(x) = 0.
To find the critical points, we need to find where the derivative of g(x), g'(x), is equal to zero.
g(x) = 1/8x^2 - 4�‾(x)
To find g'(x), we can differentiate g(x) with respect to x.
g'(x) = d/dx(1/8x^2) - d/dx(4�‾(x))
The derivative of x^n is nx^(n-1), so:
g'(x) = (1/8) * d/dx(x^2) - 4�‾'(x)
g'(x) = (1/8) * 2x - 4�‾'(x)
Simplifying further, we have:
g'(x) = x/4 - 4�‾'(x)
To find when g'(x) = 0, we set it equal to zero:
x/4 - 4�‾'(x) = 0
Since 4�‾'(x) = 4(1) = 4, we can rewrite the equation as:
x/4 - 4 = 0
x/4 = 4
x = 16
So, the critical point is x = 16.
Now let's move on to step 2.
Step 2: Determine the endpoints of the interval [0, 49].
The given interval is [0, 49], so the endpoints are x = 0 and x = 49.
Now, let's proceed to step 3.
Step 3: Evaluate the function at the critical point and endpoints.
We need to evaluate the function g(x) at x = 0, x = 16, and x = 49.
g(0) = (1/8)(0)^2 - 4�‾(0) = 0 - 4(0) = 0
g(16) = (1/8)(16)^2 - 4�‾(16) = 32 - 4(4) = 32 - 16 = 16
g(49) = (1/8)(49)^2 - 4�‾(49) = 306.125
Now we can move on to step 4.
Step 4: Compare the values obtained to find the absolute maximum and minimum.
Comparing the values obtained, we find:
Absolute maximum value: g(49) = 306.125
Absolute minimum value: g(0) = 0
Therefore, the absolute maximum value of the function g(x) on the interval [0, 49] is 306.125, and the absolute minimum value is 0.