# What is the range of the function -x^2 + 4x -10?

Apr 28, 2017

$\left(- \infty , - 6\right]$

#### Explanation:

$f \left(x\right) = - {x}^{2} + 4 x - 10$

Since the coefficient of ${x}^{2}$ is negative, the quadratic function, fx) will have a maximum value.

$f ' \left(x\right) = - 2 x + 4$

$\therefore f \left(x\right)$ will have a maximum value where: $- 2 x + 4 = 0$

$2 x = 4 \to x = 2$

$\therefore {f}_{\text{max}} = f \left(2\right) = - 4 + 8 - 10 = - 6$

$f \left(x\right)$ has no lower bound.

Hence the range of $f \left(x\right)$ is $\left(- \infty , - 6\right]$

This can be seen from the graph of #f(x) below.

graph{-x^2+4x-10 [-37.43, 44.77, -32.54, 8.58]}