Sketch the function
$\displaystyle f(x)=\min_{t\le x} t^2$ for all real $x$.
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What is the minimum value of
$t^2$when$t$varies between$-\infty$and$-3?$ -
How about between
$-\infty$and$-1?$ -
How about between
$-\infty$and$+1?$
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We are looking for the minimum value of
$t^2$for$t\in(-\infty,x]$. This is$x^2$when$x<0$, and$0$when$x\ge0$.