# Practice Paper 2 Question 1

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

## Hints

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