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All values at which $f$ has a local minimum.

and

All local minimum values of $f$.

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4 Answers

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Yet another way to repeat the previous answers: if the point $(x,y)$ on the graph of $f$ corresponds to a local minimum, then one says that this local minimum value "is $y$" and that it "occurs at $x$".

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The first one refers to the domain points, the second one refers to refers to the value your function assumes in those points.

Namely: take $f(x)=\sin x:\;\;x=-\pi/2+2k\pi$ are the values in which $f$ attains its minimum (in this case it's global, but it doesn't matter); then $f(-\pi/2+2k\pi)=-1$ it's the minimum.

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The first is asking for all points $x$ such that $f(x)$ is at a local minimum, and the second is asking for all values $y$ such that there exists $x$ with $f(x)=y$, a local minimum.

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In the first case, you're asking for $p$ such that $f(x)$ has a local minimum at $x=p$. In the second, you're asking for $f(p)$ such that $f(x)$ has a local minimum at $x=p$.

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