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Proving something is a local minimum
I have a question about the following problem:
If $g$ is twice continuously differentiable in $[0, 1]$ and $g'(0) = g'(1) = 0$ and the quadratic polynomial $g(x)$ has only $2$ points of local minima in $[0, 1]$, show that $g(x)$ is a global minimum.
I am really not sure what to do here because the question goes on from there and we are to show that g is $C^\infty$ in $[0, 1]$ and that all local minima are global minima. We are also told that $g'(x)$ exists for all $x \in [0,1]$. I really don’t know where to begin…
A:
HINT:
Sketch the graph. Note that for all $x$ in $[0,1]$, $$g(x)-g(0)=g(1)-g(0)=0.$$
Therefore, $g(x)\ge0$ for all $x$ in $[0,1]$.
Q:
How to capture a regex starting with an end of line without triggering multiline?
How to capture a regex starting with an end of line without triggering multiline?
The following line:
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