So i have this really important assignment, were i have made a couple of statements and I want to check with you if they are correct or not :
A function is continuous at a point xo, but is not differentiable at xo. In an interval around xo, it is twice differentiable and f'' does not change the sign. Then this implies that x0 is a corner point.
By corner point i mean , for example, the point where x=0 in a function like f=absolute value of x.
My second statement that i wanted to check is (reguarding linear algebra):
There are five linearly independent vectors. These five linearly independent vectors must have at least 5 components each .
Thanks for the help :) really appreciate it
Both statements are true, I'm not sure if the first one requires the function to be twice differentiable.
For the first one, if the function is continuous, but not differentiable that means that there are no holes, nor-discontinuities, so there must be a kink (not real terminology I don't think). If it's twice differentiable in a neighborhood around $x_0$, then you only have the kink at $x_0$, so it is similar to an absolute value.
For the second one, you can combine your vectors $v = {v_1, v_2, v_3, v_4, v_5}$ into a 5xn matrix. If $n\leq 4$, then there cannot be a pivot in every column, which implies the vector set is dependent. So a set of 5 independent vectors must have at least 5 elements