Quadratic Optimisation

The process of finding the maximum or minimum value of a functions is called optimisation.
For the quadratic function $y=ax^2+bx+c$, we have already seen that the vertex has $x$-coordinate $-\dfrac{b}{2a}$.

$a>0$: the minimum value of $y$ occurs at $x=-\dfrac{b}{2a}$ Quadratic Optimisation - Minimum $a<0$: the maximum value of $y$ occurs at $x=-\dfrac{b}{2a}$ Quadratic Optimisation - Maximum

Example 1

Find the maximum or minimum value of $y=x^2+6x-1$ and the corresponding value of $x$.

Example 2

Find the maximum or minimum value of $y=-2x^2+8x+1$ and the corresponding value of $x$.

Example 3

The profit in selling $x$ computers per day, is given $P=-3x^2 + 120x - 400$ dollars. Find the maximum profit per day.