can a parabolic curve have two turning points? the curve $v= 0.00005(t-200)^2 - 1$ seems to have only $1$ minimum point to me at $(200,-1)$ as a minimum point.
(i completed the squares to find it)
but in my book it says it also has a maximum point at $(0,1)$
$0\leq t\leq800$ (inclusive)
my question is how would you have found the maximum point? i learned the trigonometric graphs of sine and cosine and know that they have multiple turning points but how can a quadratic graph have multiple turning points?
could anyone please  explain why?
When looking for a curve's maxima or minima, you need to examine two things - inflection points and discontinuities. Inflection points are the turning points that you seem to be familiar with, which can be found through the typical use of derivatives. Derivatives don't exist at discontinuous points, however, so you need to check them individually. This parabolic function is discontinuous at the endpoints of t=0 and t=800, so you need to check values there. Basically, you can have maxima/minima at points where the slope is not equal to zero - these aren't &quot;turning points&quot;, but rather endpoints or some other kind of discontinuity. You can see, for example, that the straight line x=y defined for [0,10] has no inflection points at all, but still has a maximum and minimum.
When looking for a curve's maxima or minima, you need to examine two things - inflection points and discontinuities. Inflection points are the turning points that you seem to be familiar with, which can be found through the typical use of derivatives. Derivatives don't exist at discontinuous points, however, so you need to check them individually. This parabolic function is discontinuous at the endpoints of t=0 and t=800, so you need to check values there. Basically, you can have maxima/minima at points where the slope is not equal to zero - these aren't &quot;turning points&quot;, but rather endpoints or some other kind of discontinuity. You can see, for example, that the straight line x=y defined for [0,10] has no inflection points at all, but still has a maximum and minimum.