# What are the critical values, if any, of # f(x)=cscxtanx-sqrt(xcosx) in [0,2pi]#?

By signing up, you agree to our Terms of Service and Privacy Policy

To find the critical values of ( f(x) = \csc(x) \tan(x) - \sqrt{x \cos(x)} ) in the interval ([0, 2\pi]), we need to find the values of ( x ) where the derivative of ( f(x) ) is equal to zero or undefined.

The derivative of ( f(x) ) with respect to ( x ) is:

( f'(x) = -\csc(x) \cot(x) - \frac{\sin(x) + x\sin(x) - 2\cos(x)}{2\sqrt{x \cos(x)}} )

To find where ( f'(x) ) is undefined, we look for values of ( x ) where ( x \cos(x) = 0 ). This occurs at ( x = 0, \pi, 2\pi ).

To find where ( f'(x) ) is zero, we solve the equation:

( -\csc(x) \cot(x) - \frac{\sin(x) + x\sin(x) - 2\cos(x)}{2\sqrt{x \cos(x)}} = 0 )

This equation has no simple algebraic solution. We would typically use numerical methods to approximate the solutions. These critical values would be within the interval ([0, 2\pi]).

So, the critical values of ( f(x) ) in the interval ([0, 2\pi]) are ( x = 0, \pi, 2\pi ), and possibly other values found through numerical methods.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- Is #f(x)=1/x-1/x^3+1/x^5# increasing or decreasing at #x=1#?
- Given the function #f(x)=x^3-9x^2+24x-18#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [2,4] and find the c?
- Given the function #f(x)=(x^2-1)/(x-2)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-1,3] and find the c?
- How do use the first derivative test to determine the local extrema #x^2+1#?
- How do you find the critical points and local max and min for #y=(x-1)^4#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7