Is #f(x)=e^x/cosx-e^x/sinx# increasing or decreasing at #x=pi/6#?

Answer 1

#f(x)# is increasing at #pi/6#.

To figure out whether a function is increasing or decreasing at a certain point, we can take the function's derivative. If the derivative is positive, the function is increasing at that point, and if the derivative is negative, the function is decreasing at that point. If we take the derivative of this function using the quotient rule, we get #f'(x)=e^x(cos(x)sin^2(x)+sin^3(x)-sin(x)cos^2(x)+cos^3(x))#. Plugging in #pi/6#, we find that the derivative at that point #~~1.04#, so we know that the function is increasing.
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) = \frac{e^x}{\cos x} - \frac{e^x}{\sin x} ) is increasing or decreasing at ( x = \frac{\pi}{6} ), we need to evaluate the derivative of ( f(x) ) and then substitute ( x = \frac{\pi}{6} ).

The derivative of ( f(x) ) with respect to ( x ) is given by: [ f'(x) = \frac{e^x}{\cos x} + e^x \tan x - \frac{e^x}{\sin x} - e^x \cot x ]

Now, substitute ( x = \frac{\pi}{6} ) into ( f'(x) ) to find the value of the derivative at ( x = \frac{\pi}{6} ): [ f'\left(\frac{\pi}{6}\right) = \frac{e^{\frac{\pi}{6}}}{\cos\left(\frac{\pi}{6}\right)} + e^{\frac{\pi}{6}} \tan\left(\frac{\pi}{6}\right) - \frac{e^{\frac{\pi}{6}}}{\sin\left(\frac{\pi}{6}\right)} - e^{\frac{\pi}{6}} \cot\left(\frac{\pi}{6}\right) ]

After evaluating this expression, we can determine whether ( f(x) ) is increasing or decreasing at ( x = \frac{\pi}{6} ) by analyzing the sign of ( f'\left(\frac{\pi}{6}\right) ). If ( f'\left(\frac{\pi}{6}\right) > 0 ), then ( f(x) ) is increasing at ( x = \frac{\pi}{6} ), and if ( f'\left(\frac{\pi}{6}\right) < 0 ), then ( f(x) ) is decreasing at ( x = \frac{\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 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