Is #f(x)=cosx+cotx*sinx# increasing or decreasing at #x=pi/6#?

Answer 1

#f darr# at #x=pi/6#.

# f'(a)<0 rArr f darr# at #x=a#.
#f(x)=cosx+cotx*sinx=cosx+cosx=2cosx#
#rArr f'(x)=-2sinx rArr f'(pi/6)=-2*sin(pi/6)=-2*1/2=-1<0#.
Hence, #f darr# at #x=pi/6#.
Sign up to view the whole answer

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

Sign up with email
Answer 2

To determine if ( f(x) = \cos(x) + \cot(x) \sin(x) ) is increasing or decreasing at ( x = \frac{\pi}{6} ), we need to find the derivative of ( f(x) ) and then evaluate it at ( x = \frac{\pi}{6} ).

The derivative of ( \cos(x) ) is ( -\sin(x) ), the derivative of ( \cot(x) ) is ( -\csc^2(x) ), and the derivative of ( \sin(x) ) is ( \cos(x) ).

Therefore, the derivative of ( f(x) ) is ( -\sin(x) - \cot(x)\csc^2(x) ).

Evaluating this derivative at ( x = \frac{\pi}{6} ), we get ( -\frac{\sqrt{3}}{2} - \frac{2\sqrt{3}}{3} = -\frac{5\sqrt{3}}{6} ), which is negative.

Since the derivative is negative at ( x = \frac{\pi}{6} ), the function is decreasing at that point.

Sign up to view the whole answer

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

Sign up with email
Answer from HIX Tutor

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.

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7