Q1) For a continuous function $f$ whose domain is all real numbers, if $f'(p)$ is undefined, then $x = p$ could be a local maximum or minimum of $f$.
I say that at corners it could be maxima or minima but if we are taking the derivative it will be considered undefined and the maxima and minima points has to be zero. (False)
Q2) For a function $f$ whose domain is all real numbers, if $f''(p) = 0$, then the graph of $f$ has an inflection point at $x = p$.
I would say it is true, I saw some examples with this definition occurs but I am not sure. ( True)
Q3) For a function $f$ whose domain is all real numbers, if $f'(p)=0$, then $f(x)$ has a local minimum or local maximum at $x = p$.
It is true for sure, but does it only have to be local maxima or minima ? because sometimes it crosses the $x$ axis while not having maxima or minima. Am I right ? (True)
Q4) For a function $f$ whose domain is all real numbers, a local maximum of $f$ only occurs at a point where $f'(x)=0$.
That has the same issue as the first question and I would say true. (True)
Best Answer
Consider:
Q1) $f(x) = |x|$
Q2) $f(x) = x^4$
Q3) $f(x) = x^3$
Q4) $f(x) = -|x|$
at $x = 0$.
Let me know what your T/F answers are after you think about these examples.