How do I find the extreme values for the function y=sinx+cosx on the interval 0<x<2(pi)
get the derivative to get the maximum/minimum:
y = sinx + cosx
y' = 0 = cosx - sinx
cos x = sin x
x=pi/4 *the value of x in which y is max
x=-(pi/4) *the value of x in which y is min
you can actually check this by graphing the equation,,
so there,, =)
To find the extreme values of a function on a given interval, you need to follow these steps:
1. Find the derivative of the function.
In this case, the derivative of y = sin(x) + cos(x) can be found by applying the derivative rules. The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). So, the derivative of y with respect to x is dy/dx = cos(x) - sin(x).
2. Determine the critical points.
To find the critical points, set the derivative equal to zero and solve for x. So, cos(x) - sin(x) = 0.
To simplify the equation, you can rewrite sin(x) as cos(π/2 - x) using the identity sin(x) = cos(π/2 - x).
So, cos(x) - sin(x) = cos(x) - cos(π/2 - x) = 0.
Combining like terms, you get: cos(x) - cos(π/2 - x) = 0.
Using the cosine difference formula: cos(a) - cos(b) = -2sin((a + b)/2)sin((a - b)/2), the equation becomes: -2sin((x + π/2 - x)/2)sin((x + π/2 + x)/2) = 0.
Simplifying further, you get: -2sin(π/4)sin(2x + π/4) = 0.
Since sin(π/4) is a non-zero constant, the equation simplifies to: sin(2x + π/4) = 0.
Solve for x by setting 2x + π/4 = kπ, where k is an integer.
So, 2x = kπ - π/4.
Simplifying further, you get: x = (kπ - π/4)/2.
Since the interval is 0 < x < 2π, you need to find the values of x that satisfy this condition.
3. Evaluate the function at the critical points.
Evaluate the function y = sin(x) + cos(x) at the critical points found in step 2. That is, substitute the values of x into the function and calculate y.
For example, for x = (kπ - π/4)/2, substitute this value into the function y = sin(x) + cos(x) and calculate the corresponding y-values.
4. Find the maximum and minimum values.
Compare the y-values calculated in step 3 to determine which are the maximum and minimum values of the function on the given interval 0 < x < 2π.
Note: Since the given function y = sin(x) + cos(x) is periodic with a period of 2π, it may have multiple maximum and minimum values on the interval 0 < x < 2π.