Question: Without looking at the graph, how would I know if the value
at a critical point of a function is a local extrema or a global
extrema?
To give details as to what kind of confusion I am having, consider the function $$f(x)=x^{2/3}(x^2 – 4).$$
We find that $$f'(x)=\frac{8x^2-8}{3\sqrt[3]{x}},$$
so that the critical points are $-1, 0, 1$. Then we have that $f(\pm1) = -3$ and $f(0)=0$. Here $-3$ is the global min value of the function, but $0$ is the local min value of the function. I was able to tell all of this by looking at the graph, so my question becomes: can I tell the difference b/w local and global max/min values without plotting the function?
Best Answer
When talking about a real function $f(x) = y$ all extremes of the function are either critical points, boundary points or points which diverge to infinity, so all you need to do is to look at the value of the function at these points to determine the maximum and the minimum of the function. Here you need to consider $$\lim_{x\to\pm\infty}f(x) $$ as well. i think that's what confused you here, always consider the boundaries of the function you're looking at, which in the case of this function is $\pm\infty$ as it is defined for all real numbers.