Find absoulute maximum and absolute minimum of the function f(x)=2cost+sin2t
To find the absolute maximum and absolute minimum of the function f(x) = 2cos(t) + sin(2t), we need to follow these steps:
Step 1: Find the critical points of the function by finding where the derivative of f(x) equals zero or is undefined.
Step 2: Evaluate the function at the critical points and the endpoints of the interval to determine the maximum and minimum values.
Step 3: Compare the function values at these points to find the absolute maximum and absolute minimum.
Let's go through each step in detail:
Step 1: Find the critical points of f(x).
To find the critical points, we need to find where the derivative of f(x) equals zero or is undefined. The derivative of f(x) can be found by applying the chain rule.
f'(x) = -2sin(t) + 2cos(2t) * (2)
Setting f'(x) equal to zero:
0 = -2sin(t) + 4cos(2t)
Now we need to solve this equation for t. However, it might not be possible to find an exact solution analytically.
Step 2: Evaluate the function at the critical points and endpoints.
In this case, since we were not able to find any analytical solutions for t, we can use numerical methods or graphing calculators to estimate the values of t at which f(x) is at a maximum or minimum.
Step 3: Compare the function values at these points to find the absolute maximum and absolute minimum.
Once you have estimated the values of t, plug them back into the original function f(x) = 2cos(t) + sin(2t) to evaluate the function. Also, evaluate the function at the endpoints of the given interval.
Compare all the values obtained to find the largest and smallest values to determine the absolute maximum and absolute minimum, respectively.
Keep in mind that without a specific interval or range given for t, it is not possible to obtain a specific solution for the absolute maximum and absolute minimum of f(x).