True or False question about derivatives

[lauren.m]

So I'm having trouble answering a few true or false questions on this homework.
1.) If f''(x) is always positive, then the function f must have a relative minimum value.
I'm struggling with this one because I thought it would be true since f''(x) would always be positive meaning f'(x) would be increasing, so f would be always be positive but I'm not sure.
2.) If f'(2)=0 and f''(2) is less than 0, then x=2 locates a relative maximum value of f.
I think this is false because f'(x) might not switch signs, meaning it would not be a maximum point, but someone else told me it was true.

 

[Steven G]

1) Most of what you said is true. f"(x) > 0 implies that f'(x) would always be increasing. Now draw a few of these functions and decide if f(x) must have a minimum or not.
BTW, if f'(x) >0 that does not make f(x) positive. Think of this: You have a function where f'(x) >0 AND f(x)>0. If you lower the function (ie graph) enough f(x) will turn negative but f'(x) does not change. Why does raising or lowering a graph not change the derivative of f'(x)?

2) Why do you tell if you have a min at x=2 for any function f(x)?

 

[skeeter]

1) consider an exponential function like [MATH]y=e^x[/MATH]

 

[JeffM]

With respect to # 2, if a function is twice differentiable on (a, b), a < c < b, and f(c) is a local maximum or minimum, then f'(c) = 0. If f''(c) < 0, then f(c) is a relative maximum; if f''(c) > 0, then f(c) is a local minimum.

Unfortunately, if f'(c) = 0 = f''(c), f(c) may be neither a local maximum nor a local minimum.

To put it a little more crudely, if f(c) is a local extremum of a differentiable function, then f'(c) = 0. But you cannot reverse that unless f''(c) is not zero.