Find the absolute maximum and absolute minimum values of
f(x)=sinx + (�ã3) cosx on the closed interval [0,ƒÎ/3]
take the derivative, set it equal to zero and solve.
I can 't make out your interval
(I got 30º for x, take it from there)
To find the absolute maximum and minimum values of the function f(x) = sin(x) + √3 cos(x) on the closed interval [0, π/3], we need to follow these steps:
1. Take the derivative of f(x) with respect to x to find its critical points.
2. Identify the values of x that make f'(x) equal to zero or undefined.
3. Evaluate f(x) at the critical points and the endpoints of the interval [0, π/3].
4. Compare the values of f(x) at these points to determine the absolute maximum and minimum.
Let's begin:
Step 1: Find the derivative of f(x)
f'(x) = cos(x) - √3 sin(x)
Step 2: Find the critical points
To find the critical points, we set f'(x) equal to zero and solve for x:
cos(x) - √3 sin(x) = 0
To simplify this equation, we divide both sides by cos(x):
1 - √3 tan(x) = 0
Now, rearrange the equation:
tan(x) = 1/√3
x = arctan(1/√3)
Step 3: Evaluate f(x) at the critical points and endpoints
We need to evaluate f(x) at the critical points x = arctan(1/√3) and the endpoints x = 0 and x = π/3.
f(0) = sin(0) + √3 cos(0) = 0 + √3 = √3
f(π/3) = sin(π/3) + √3 cos(π/3)
= (√3/2) + (√3/2)
= √3
f(arctan(1/√3)) = sin(arctan(1/√3)) + √3 cos(arctan(1/√3))
= (1/2) + (√3/2)
= (1 + √3)/2
Step 4: Determine the absolute maximum and minimum
Comparing the values we obtained:
f(0) = √3
f(π/3) = √3
f(arctan(1/√3)) = (1 + √3)/2
Since f(0), f(π/3), and f(arctan(1/√3)) all have the same value of √3, this is the absolute maximum value on the interval [0, π/3].
There is no other point at which f(x) can be lower than √3, so there is no absolute minimum.
In summary, the absolute maximum value of f(x) = sin(x) + √3 cos(x) on the closed interval [0, π/3] is √3, and there is no absolute minimum.