Find the intervals on which sinx-(root3)(cosx) is increasing and decreasing
y = sinx - 3^0.5 * (cosx)
dy/dx = cos x + 0.5 * 1/ (3^0.5) sin x
= cos x + 0.29 sin x
where is that zero? where is it plus? where is it minus?
y = sinx - √3 cosx
y' = cosx + √3 sinx = 2sin(x + π/6)
To find the intervals on which the function sinx - √3cosx is increasing and decreasing, we'll need to find the first derivative of the function and determine where it is positive or negative.
First, let's find the derivative:
f(x) = sinx - √3cosx
f'(x) = cosx + √3sinx
Next, we'll set the derivative equal to zero and solve for x to find the critical points:
cosx + √3sinx = 0
We can rewrite this equation as:
cosx = -√3sinx
Now, divide both sides of the equation by cosx:
1 = -√3tanx
To find the critical points, we'll need to find the values of x that satisfy this equation. We know that tanx = sinx/cosx, so we can rewrite the equation as:
1 = -√3(sinx/cosx)
Multiply both sides by cosx to get rid of the denominator:
cosx = -√3sinx
Now, recall that cos²x + sin²x = 1. Substituting this identity into the equation:
1 = -√3sinx(√(1 - sin²x))
Simplifying further:
1 = -√3sinx√(1 - sin²x)
Squaring both sides of the equation:
1 = 3sin²x(1 - sin²x)
Expanding:
1 = 3sin²x - 3sin⁴x
Rearranging and factoring:
3sin⁴x - 3sin²x + 1 = 0
This is a quadratic equation in sin²x. We can solve this equation using the quadratic formula:
sin²x = [-(-3) ± √((-3)² - 4(3)(1)))] / (2(3))
Simplifying further:
sin²x = [3 ± √(9 - 12)] / 6
sin²x = [3 ± √(-3)] / 6
Since the square root of a negative number is not real, there are no real solutions for sin²x. Therefore, there are no critical points for the function f(x) = sinx - √3cosx.
Since there are no critical points, the function is either always increasing or always decreasing. To determine this, we can analyze the behavior of the derivative:
f'(x) = cosx + √3sinx
Since the cosine function swings between -1 and 1 while the sine function swings between -√3 and √3, the sum of these two functions can be negative or positive depending on the values of x.
To summarize:
- If cosx + √3sinx > 0, the function f(x) = sinx - √3cosx is increasing.
- If cosx + √3sinx < 0, the function f(x) = sinx - √3cosx is decreasing.
Since we don't know the specific range of x values provided, we cannot determine the intervals on which the function is increasing or decreasing without additional information.