How do you determine whether the given critical point x=-2.5 is the location of a maximum, a minimum, or a point of inflection for y= 2x^2 + 10x - 7?

Apr 25, 2015

The simplest way is to recognize that any equation of the form:
$y = a {x}^{2} + b x + c$
is a parabola that opens upward if $a > 0$ (the critical point is a minimum)
and
is a parabola the opens downward if $a < 0$ (the critical point is a maximum)

If the equation is not as obvious, the second derivative will tell you if the function has an increasing slope (the point is a minimum),
or a decreasing slope (the point is a maximum),
or the slope is changing from minimum to maximum (the point is a point of inflection).

Given
$y = 2 {x}^{2} + 10 x - 7$
$y ' = 2 x + 10$
$y ' ' = 2$
The slope is constantly increasing (it starts very negative, becomes less and less negative until at the critical point it becomes zero, and then continues to become more and more positive).

That is, the critical point for
$y = 2 {x}^{2} + 10 x - 7$
is a minimum.