Q&A for How to Find Inflection Points

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What if the second derivative intersects with the x-axis, but does not dip below it?

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wikiHow Staff Editor

Staff Answer

The second derivative has to cross the x-axis for there to be an inflection point. If the second derivative only touches the x-axis but doesn't cross it, there's no inflection point.

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What if the second derivative is a constant? How do I find the inflection point?

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Inflection points are where the second derivative changes sign. If it is constant, it never changes sign, so there exists no inflection point for the function.

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Can the first derivative become zero at an inflection point?

Orangejews

Community Answer

Yes, for example x^3. It changes concavity at x=0, and the first derivative is 0 there.

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Find the value of x at which maximum and minimum values of y and points of inflection occur on the curve y = 12lnx+x^2-10x.

Community Answer

Take the derivative and set it equal to zero, then solve. These are the candidate extrema. Take the second derivative and plug in your results. If it's positive, it's a min; if it's negative, it's a max. Set the second derivative to 0 and solve to find candidate inflection points. We can rule one of them out because of domain restrictions (ln x). Confirm the other by plugging in values around it and checking the sign of the second derivative.

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Why isn't y^2=x a function? Can I say that x is function of y? How do I determine the dependent and independent variable in a relation or function?

Community Answer

For that equation, it is correct to say x is a function of y, but y is not a function of x. A common notational convention is to use x for an independent variable and y for a dependent variable, and for function to mean that the dependent variable is uniquely determined by the independent variable. Saying "y^2 = x is not a function" is true if the author implicitly assumed those conventions, but it would have been better to state them explicitly to avoid any confusion.

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Why does 6x = 0 become '0' and not x = -6?

Community Answer

Multiplying 6 by -6 will give you a result of -36, not 0. Multiply a number by 0 to achieve a result of 0.

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Why do we set the both first and second derivative equal to zero to find the points?

Caeiia

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You only set the second derivative to zero. This is because an inflection point is where a graph changes from being concave to convex or vice versa. That change will be reflected in the curvature changing signs, or the second derivative changing signs. When the second derivative changes from positive to negative or negative to positive, it will at one point in time be zero. That point where it is zero is exactly when it starts to change.

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If my second derivative is 2/x, does it have an inflection point?

Community Answer

This depends on the critical numbers, ascertained from the first derivative. You test those critical numbers in the second derivative, and if you have any points where it goes from one concavity before to another after, then you have a point of inflection.

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