Minimal and Maximum Value of Expressions

To find the minimum or maximum value of an expression, calculus (finding the derivative and setting it to zero) can be a slow method. It's often faster to use algebraic manipulation, such as completing the square.

For an expression like $2(x-1)^2 + 3$, the minimum value occurs when the squared term is zero, since squares are always non-negative. The minimum value of $(x-1)^2$ is 0, so the minimum of the whole expression is $2(0)+3=3$.

For expressions with trigonometric functions, remember that the minimum value of sin and cos is -1, and the maximum is 1.