even function with negative first derivative?

Question

I'm pretty sure this scenario is impossible but want to check for sure. I have been asked to graph an even function whose first derivative is negative on the interval (-2,2). Impossible right?

Answer 1

The derivative of an even function should be an odd function.  Examples:

 

(i)   cos(x) is even, cos(x)' = -sin(x), which is odd.

 

(ii)  x⁴ + 2x² - 3 is even, (x⁴ + 2x² - 3)' = 4x³ + 4x, which is odd.

 

So if f(x) is even, f'(x) should be odd, and hence f'(x) should have opposite signs at -x and x.  It looks like f'(x) cannot be negative over the entire interval of (-2,2), since its value must change from f(x) to -f(x) when you evaluate it at -x.  I agree with you.