Subject: Calculus

Stationary Point

So what is a stationary point?


The point at the peak is a stationary point.

A point or points in the graph of a function that tells that the function itself is not decreasing nor increasing is called a stationary point. Not all functions have stationary points. An inclined line for example is a function which has no stationary point or points. A curve of a mountain viewed in two dimension has can have its stationary point at the peak (of course, because at the peak, the mountain stops increasing nor decreasing).

At the stationary points...

  • the gradient (derivative) is zero,
  • the tangent line is horizontal and
  • the function is neither increasing (positive slope) or decreasing (negative slope).

Two types of stationary points

There are two kinds of stationary points: Turning point - function changes direction at a stationary point. Turning points are either maximums or minimums - depending on the value of y at the turning point and the points nearby; Point of inflection - at this point, function's slope is diminish but there is no change in the direction of the curve. It only changes its concavity.

Okay, so how do we determine the stationary points?

Determining stationary points of a function at this point is very important because it is very useful later as we discuss the applications of the uses of stationary points such as curve sketching and optimization problems. To determine the stationary points, we just simply take the first derivative of that function and equate to zero, thus f'(x)=0. Then solve for the points.

Illustration


Stationary point of f(x)=-x^2+1 is at the point (0, 1).

Suppose we want to find the stationary points of the function f(x)=-x^2+1. First, let's take its first derivative. Thus,

f(x)=-x^2+1
f'(x)=-2x

Equating to zero and solve, we have

f'(x)=-2x=0
x=0

Then putting back x=0 to the function f(x), we have

f(x)=y=-(0)^2+1

y=1

Thus, the point (x,y)=0,1 is the stationary point of the function f(x)=-x^2+1(see figure at the right). Notice that the function behaves like a mountain or a hill and as expected, the stationary point is located at the peak.!

Stationary points and critical points

Technically, a stationary point is a critical point. Like stationary points, critical points can also be determined using the steps above. The only difference between the two is that a critical point is more general because it also includes points of singularities of the derivative of that function. So when the derivative of a function is undefined at a certain point, then that point is said to be a critical point but not necessarily a stationary point.

Example #1

Find the stationary point/s of the function y=x^2+2x+2.

Answer: (1, -1)

Example #2

Solve for the stationary points of the following function: y=x^2y+2.

Answer: (2, 0)

Example #3

What are the stationary point/s of the function f(x)=mx+b.

Answer: A line has no stationary points!.

Example #4

Does circles have stationary point/s?

Answer: No.

Example #5

Solve for the stationary points of the function y=\sin{x}.

Answer: (1, \frac{n\pi}{2})

where n stands for natural numbers and in this case, includes the negative ones.

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