f(x)= −10.4sin(3x)+8.5cos(3x) find max and min on interval ( -0.81, 1.27 )
To find the maximum and minimum values of the function f(x) = -10.4sin(3x) + 8.5cos(3x) on the interval (-0.81, 1.27), we will need to locate the critical points and endpoints of the interval.
1. Critical points:
To find the critical points, we need to calculate the derivative of f(x) and set it equal to zero. This is because the critical points occur where the slope of the function is zero or undefined.
Let's start by finding the derivative of f(x) with respect to x:
f'(x) = -10.4 * d(sin(3x))/dx + 8.5 * d(cos(3x))/dx
Using the chain rule, we can differentiate sin(3x) and cos(3x) as follows:
f'(x) = -10.4 * 3 * cos(3x) + 8.5 * (-3) * sin(3x)
Simplifying further, we have:
f'(x) = -31.2cos(3x) - 25.5sin(3x)
Now, let's solve for f'(x) = 0:
-31.2cos(3x) - 25.5sin(3x) = 0
We can rearrange this equation to separate the cosine and sine terms:
-25.5sin(3x) = 31.2cos(3x)
Dividing both sides by cos(3x), we get:
tan(3x) = -25.5/31.2
Now, we can use the inverse tangent function to solve for x:
3x = arctan(-25.5/31.2)
x = arctan(-25.5/31.2) / 3
Using a calculator, we find the value of x to be approximately -0.347.
2. Endpoints:
Next, we need to evaluate f(x) at the endpoints of the given interval.
For x = -0.81:
f(-0.81) = -10.4sin(3(-0.81)) + 8.5cos(3(-0.81))
Calculate the value using a calculator: f(-0.81) ≈ 7.177
For x = 1.27:
f(1.27) = -10.4sin(3(1.27)) + 8.5cos(3(1.27))
Calculate the value using a calculator: f(1.27) ≈ -10.324
3. Maximum and Minimum:
Now, we compare the function values at the critical point and endpoints to determine the maximum and minimum.
The function value at the critical point, x = -0.347, is:
f(-0.347) = -10.4sin(3(-0.347)) + 8.5cos(3(-0.347))
Calculate the value using a calculator: f(-0.347) ≈ 10.843
Comparing the function values, we can see that:
The maximum value on the interval (-0.81, 1.27) is approximately 10.843, which occurs at x = -0.347.
The minimum value on the interval (-0.81, 1.27) is approximately -10.324, which occurs at x = 1.27.
Therefore, the maximum and minimum values of f(x) on the interval (-0.81, 1.27) are approximately 10.843 and -10.324, respectively.