Stationary point

In calculus, a stationary point is a point at which the slope of a function is zero. Stationary points can be found by taking the derivative and setting it to equal zero. For example, to find the stationary points of

$f(x) = x^3 + 3x^2 + 3x + 4$

one would take the derivative:

$f'(x) = 3x^2 + 6x + 3$

and set this to equal zero.

$3x^2 + 6x + 3 = 0$
$x^2 + 2x + 1 = 0$
$(x + 3)(x + 1) = 0$
$x + 1 = 0$
$x=-1$

This gives the x-value of the stationary point. To find the point on the function, simply substitute this value for x in the original function.

$f(x) = x^3 + 3x^2 + 3x + 4$
$f(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 4$
$f(-1) = -1 + 3 - 3 + 4$
$f(-1) = 3$

So the coordinates for the stationary point would be $(-1, 3)$.

One can then use this to find if it is a minimum point, maximum point or point of inflection.

This can be done by further differentiating the derivative and then substituting the x-value in. If the calculation results in a value less than 0, it is a maximum point. If the calculation results in a value greater than 0, it is a minimum point. If the calculation is equal to 0, it is a point of inflection.