find the absolute extrema of f(x)=sin pi x on [-1,2]
Not sure how to do this problem....
I read you equation as
f(x) = sin (πx)
f'(x) = πcos(πx)
for local max/min, f'(x) = 0
cos(πx) = 0
πx = -π/2 or πx = π/2 or πx= 3π/2
x = ± 1/2 or x = 3/2 , but our domain is [-1,2]
so x = -1/2 or x = 1/2
so
f(-1/2) = sin (-π/2) = -1
f(1/2) = sin (π/2) = 1
also look at endpoints of domain
f(-1) = sin(-π) = 0
f(2) = sin(2π) = 0
so the max is 1 and the min is -1
To find the absolute extrema of the function f(x) = sin(pi x) on the interval [-1, 2], we need to find the maximum and minimum values of the function within that interval.
1. First, we need to find the critical points of f(x) on the interval [-1, 2]. Critical points occur when the derivative of the function is equal to zero or does not exist.
The derivative of f(x) = sin(pi x) is f'(x) = pi*cos(pi x).
Setting f'(x) = 0, we get:
pi*cos(pi x) = 0
This equation is true when cos(pi x) = 0. The values of x for which cos(pi x) = 0 are x = -1/2, 0, and 1/2. However, only x = -1/2 and 1/2 are within the interval [-1, 2].
2. We now need to evaluate the function f(x) at the critical points and the endpoints of the interval [-1, 2].
f(-1) = sin(pi * (-1)) = sin(-pi) = 0
f(-1/2) = sin(pi * (-1/2)) = sin(-pi/2) = -1
f(1/2) = sin(pi * (1/2)) = sin(pi/2) = 1
f(2) = sin(pi * (2)) = sin(2pi) = 0
3. Now, we compare the function values at the critical points and the endpoints to determine the maximum and minimum values.
The maximum value of f(x) on the interval [-1, 2] is 1, which occurs at x = 1/2.
The minimum value of f(x) on the interval [-1, 2] is -1, which occurs at x = -1/2.
Therefore, the absolute maximum value of f(x) = sin(pi x) on the interval [-1, 2] is 1, and it occurs at x = 1/2.
The absolute minimum value of f(x) = sin(pi x) on the interval [-1, 2] is -1, and it occurs at x = -1/2.
To find the absolute extrema of a function, you need to determine the maximum and minimum values of the function over a given interval. In this case, we are given the function f(x) = sin(pi x) and the interval [-1, 2].
To find the extremum (maximum/minimum) points, we need to follow these steps:
Step 1: Find the critical points of the function within the given interval. Critical points occur where the derivative of the function is either zero or undefined. Since f(x) = sin(pi x), we need to find where the derivative of f(x) is zero or undefined.
The derivative of f(x) is f'(x) = pi * cos(pi x). Setting this derivative to zero, we get:
pi * cos(pi x) = 0
Since cos(pi x) can only be zero at x = -1/2, 1/2, 3/2, etc., excluding the endpoints of the interval, we find that there is only one critical point within the interval [-1, 2], which is x = 1/2.
Step 2: Evaluate the function at the critical points and the endpoints of the interval. Plug in the critical points and the endpoints of the interval into the function f(x) = sin(pi x) and calculate the corresponding function values.
f(-1) = sin(pi * (-1)) = -sin(pi) = 0
f(2) = sin(pi * 2) = 0
f(1/2) = sin(pi * 1/2) = sin(pi/2) = 1
Step 3: Compare the function values obtained in step 2. The highest value is the maximum, and the lowest value is the minimum.
The function values obtained are:
f(-1) = 0
f(2) = 0
f(1/2) = 1
The maximum of the function is 1, and the minimum is 0.
Therefore, the absolute maximum value is 1, occurring at x = 1/2 within the interval [-1, 2], and the absolute minimum is 0, occurring at x = -1 and x = 2 (the endpoints of the interval).