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

1 Answer
R. Nguyen
Apr 6, 2018

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

Explanation:

The first derivative provides the slope at each point of our function. A positive slope means the function is increasing and that a negative slope means that the function is decreasing.

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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