Home > Calculus > Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)

Is #f(x)=cotx*tanx# increasing or decreasing at #x=pi/3#?

Answer 1

Neither. #f(x)# is a constant function on its domain.

#cotx = 1/tanx#, so for every #x# for which both #tanx# and #cotx# are defined, we have #cotx tanx = 1#

So for all #x# except #pi/2k# where #k# is an integer, we get #f(x)=1# so #f# is neither increasing nor decreasing on any interval.

I don't know what definition you've been given for #f# is increasing at a point, but I cannot imagine a definition that would allow a constant function to be increasing at any point.

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Answer 2

Isaiah Allen

To determine whether ( f(x) = \cot(x) \cdot \tan(x) ) is increasing or decreasing at ( x = \frac{\pi}{3} ), we can analyze the derivative of ( f(x) ) at that point.

( f'(x) = -\csc^2(x) )

Evaluate ( f'(\frac{\pi}{3}) ):

( f'(\frac{\pi}{3}) = -\csc^2(\frac{\pi}{3}) = -\frac{4}{3} )

Since ( f'(\frac{\pi}{3}) ) is negative, ( f(x) ) is decreasing at ( x = \frac{\pi}{3} ).

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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.