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How do you find all critical point and determine the min, max and inflection given #D(r)=-r^2-2r+8#?

Answer 1

Critical points #r_j# make the derivative #D'(r_j)=0#. We them have to determine whether they are maxima, minima or inflexion.

The first derivative is #D'(r)= -2r-2#, so we find the critical points by solving #-2r-2=0#, which only solution is #r=-1#. We then calculate the second derivative in #r=-1#; the second derivative is #D''=-2#, which is negative everywhere, so it is negative in the critical point #r=-1#. The function therefore has a maximum at the point #r=-1#.

Please observe that this is coherent, as the graph is a parabola pointing downwards, so it has a single maximum

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.