Re: y = x^2 stationary points
Hello, americo74!
y=x2.
The stationary points occur when 2x = 0, ie. when x = 0.
Is this a maximum or a minimum or what?
Are you familiar with the Second Derivative Test?
The second derivative is:
y′′=2 which is always positive.
Hence, the graph is always concave up:
∪
Therefore, the stationary point
(0,0) is a
minimum.
A more primitive approach . . .
At
x=0, the derivative is 0 . . . the
slope is 0.
To the left at, say, \(\displaystyle x\,=\,-1:\;y'\,=\.-2\) . . . the graph is decreasing:
↘
To the right at, say,
x=1:y′=+2 . . . the graph is increasing:
↗
Near
x=0, the graph looks like this:
↘→↗
Therefore, the stationary point must be a
minimum.