Give an example of f prime being zero but f prime prime not being zero.

Question

The question asks to give an example of a function for which f'(0)=0, but f''(0) doesn't equal zero.

 

I cant think of one because if f' is zero then wouldn't f'' always be zero as well?

Answer 1

Here is another one:

 

f(x) = x²

 

f'(x) = 2x ⇒ f'(0) = 0

 

f''(x) =  2 ≠ 0 everywhere

 

When f'(a) = 0, it just means that the tangent line at a is horizontal.  But f''(a) could be positive, as in a concave up curve like y = x², or negative, as in a concave down curve like y = -x².

Answer 2

f(x) = cos x