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# Practice Paper 2 Question 1

Sketch the function $$\displaystyle f(x)=\min_{t\le x} t^2$$ for all real $$x$$.

## Hints

• Hint 1
What is the minimum value of $$t^2$$ when $$t$$ varies between $$-\infty$$ and $$-3?$$
• Hint 2
How about between $$-\infty$$ and $$-1?$$
• Hint 3
How about between $$-\infty$$ and $$+1?$$

## Solution

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 $$x\ge0$$.

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