Locate the absolute extrema of the function on the closed interval:
y= 4/x+tan(pix/8), [1,2]
dy/dx = -4/x^2 + sec^2 (πx/8) (π/8)
= 0 for a max/min
π/(8cos^2 (πx/8)) = 4/x^2
8cos^2 (πx/8) /π = x^2/4
cos^2 (πx/8) = πx^2/32
there is no simple way to solve,
I ran it through Wolfram's amazing equation solver and got
x = ± 2.13448
http://www.wolframalpha.com/input/?i=cos%5E2+%28πx%2F8%29+%3D+πx%5E2%2F32
arrghhh, after all that, our solution is ouside the domain, so simply
if x=1
y = 4/1 + tan(π/8) = 4.414
if x = 2
y = 4/2 + tan(π/4) = 2+1 = 3
look at the graph of
http://www.wolframalpha.com/input/?i=y+%3D+4%2Fx+%2B+tan%28πx%2F8%29
everything makes sense.
To locate the absolute extrema of the function on the closed interval [1, 2], follow these steps:
Step 1: Find the critical points of the function.
To find the critical points, we need to find where the derivative of the function is zero or undefined. So, let's find the derivative of the function first.
y = 4/x + tan(πx/8)
Taking the derivative of the function with respect to x:
dy/dx = -4/x^2 + (π/8)sec^2(πx/8)
Set the derivative equal to zero to find critical points:
-4/x^2 + (π/8)sec^2(πx/8) = 0
Multiplying through by x^2 to eliminate the fraction and the denominator:
-4 + (π/8)x^2sec^2(πx/8) = 0
Next, solve for x.
Step 2: Test the critical points and the endpoints of the interval in the original function.
Now we have to test the critical points and the endpoints of the interval [1, 2] in the original function y = 4/x + tan(πx/8).
- Evaluate y at the critical points.
- Evaluate y at the endpoints x = 1 and x = 2.
Step 3: Determine the maximum and minimum values.
- Compare the values obtained from step 2 and determine the maximum and minimum values.
To locate the absolute extrema of a function on a closed interval, you need to follow these steps:
1. Find the critical points of the function within the interval by taking the derivative of the function and setting it equal to zero.
2. Evaluate the function at the critical points as well as the endpoints of the closed interval.
3. Compare the values obtained in step 2 to determine the absolute extrema.
Let's apply these steps to the given function y = 4/x + tan(px/8) on the closed interval [1, 2].
Step 1: Find the critical points
To find the critical points, we need to find the derivative of the function.
First, take the derivative of 4/x with respect to x:
(dy/dx) = -4/x^2
Next, take the derivative of tan(px/8) with respect to x using the chain rule:
(dy/dx) = p/(8cos^2(px/8))
Set the derivatives equal to zero and solve for x:
-4/x^2 + p/(8cos^2(px/8)) = 0
Step 2: Evaluate the function
Evaluate the function y = 4/x + tan(px/8) at the critical points and endpoints.
Evaluate at x = 1:
y = 4/1 + tan(p/8)
Evaluate at x = 2:
y = 4/2 + tan(2p/8) = 2 + tan(p/4)
Step 3: Compare values
Compare the values obtained in Step 2 to determine the absolute extrema.
To find the absolute minimum and maximum, compare the values at the critical points and endpoints. In this case, we need to find the minimum and maximum values of y on the interval [1, 2].
However, since p is not specified in the question, we cannot determine the exact values of the critical points and endpoints. The process described above can be used to find the critical points, and then you can substitute the value of p to get the specific critical points and evaluate them along with the endpoints.
Once you have the values of the critical points and endpoints, compare them to find the absolute extrema. Remember that the absolute extrema are the highest and lowest values the function reaches on the given interval.