Review question

# Where does $f(x) = x^2 - 2px + 3$ have its minimum? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5732

## Question

In this question we shall consider the function $f(x)$ defined by $f(x) = x^2 - 2px + 3$ where $p$ is a constant.

1. Show that the function $f(x)$ has one stationary value in the range $0< x <1$ if $0< p <1$, and no stationary values in that range otherwise.

In the remainder of the question we shall be interested in the smallest value attained by $f(x)$ in the range $0 \le x \le 1$. Of course, this value, which we shall call $m$, will depend on $p$.

1. Show that if $p \ge 1$ then $m=4-2p$.

2. What is the value of $m$ if $p \le 0$?

3. Obtain a formula for $m$ in terms of $p$, valid for $0 < p < 1$.

4. Using the axes opposite, sketch the graph of $m$ as a function of $p$ in the range $-2 \le p \le 2$.