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

Is #f(x)=cos(-x)# increasing or decreasing at #x=0#?

Answer 1

#f(x)=cos(-x)# is neither increasing nor decreasing at #x=0#

Every point on our function is sloped by the first derivative; a positive slope indicates an increasing function, and a negative slope indicates a decreasing function.

Knowing this, we can find out if #cos(-x)# is increasing or decreasing at #color(red)(x=0)# by evaluating its first derivative at that point.

Before starting, we may want to simplify our function. Because #cos(x)# is an even function, we know that #cos(-x)=cos(x)#.

#d/dx[cos(x)]#

#= -sin(x)#

Evaluate at #color(red)(x=0)#:

#-sin(color(red)0)#

#= -0#

#= 0#

In this case, the slope is neither negative or positive, but 0. Hence, #f(x)=cos(-x)# is neither increasing nor decreasing at #color(red)(x=0)#.

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Answer 2

Christopher Agresta

The function ( f(x) = \cos(-x) ) is increasing at ( x = 0 ).

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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.