what is concavity when second derivative is 2?

Question

the second derivative tests for concavity but what about when you cannot solve for x because the second derivative is a constant

Answer 1

Is that constant positive? If so, then it is convex.

Is it negative? If so, then it is concave.

Is it zero? If so, then it is a straight line.

If it is 2 -- then it is convex.

Answer 2

The second derivitive tests for inflection. If viewed with the original function, inflection is where the rate of increase or decrease in slope starts "going the other way". With simple quadratics, the inflection occurs at the minimum or maximum because that it where the slope changes from positive to negative. With the trig basic sine function, a second derivitive inflection occurs every pi radians.

Answer 3

df/dx = -2sinx-2sin2x d^2f/dx^2= -2cosx -4 sin2x = -2( cosx + 8 sinxcosx) = -2 cosx( 1+ 8sinx) D"f(x) =0 , 0