Emily A.

asked • 12/08/19

If I have a cusp on a graph of f prime (at x=3), does that mean I will have an inflection point on the graph of f (at x=3)?

John M.

tutor
No it means that is local minimum or maximum. The second derivative shows the point of inflection.
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12/09/19

1 Expert Answer

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Paul M. answered • 12/08/19

Learn "how to" do the math and why the "how to" works!

Arman G.

tutor
Yes, continuity must be assumed
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12/08/19

Arman G.

tutor
But Now thinking about it, a cusp is not differentiable but is continuous, so I’m not totally sure either
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12/08/19

Stanton D.

Right, you don't deem a point of discontinuity to be a point of inflection of a function. However, a non-differentiable point on the first derivative might arise from a "semi-cusp" -- it has a non-zero or even infinite slope (2nd derivative) on one side of the point, but a zero slope (linear or constant value original function) on the other side. Don't think that that qualifies as a cusp, per se. To make that something in the real world -- what derivative(s) of position (of your body, along a line on the floor) with respect to time would you be capable of detecting, and what common names would you call each of those derivative(s)? How about measuring, using mechanical means (gauges, levers, etc.)? F(0) = position, F(1) = velocity, F(2) = acceleration, and so on. What derivatives would a motor vehicle be capable of supplying, using the engine only? Using any other part of the car? -- Cheers, -- Mr. d.
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12/12/19

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