# If the second derivative of a function is 0 does that mean there is a point of inflection?

Update:

Sorry, the question is not clear. I meant if you input a number for x in the second derivative and if that equals 0 does that mean there is a point of inflection?

Relevance

No.

If the second derivative is zero, then the first derivative is constant, which means the function is linear.

Edit: Still no.

While it is true that the second derivative is zero at a point of inflection (A -> B), it is not true that there is a point of inflection wherever the second derivative is zero (B -> A)

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• Not always.

y=x^4

Point at 1st derivative, 2nd derivative =0.

But no inflection point.

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• it could do - the 2nd derivative indicates the rate at which the rate of change of the equation is changing

a zero indicates the rate of change is constant

Whether the rate of change becomes greater or smaller is indicated by what happens the 2nd derivative each side of the value of x when the derivative was zero

if it goes +ve it indicates the rate of change is increasing - if negative it indicates its decreasing

either way IF it changes  it indicates a change from zero to something, else,with zero indicating the point where it stopped changing and then started changing again - ie a point of inflection (And the basic equation cannot be a straight line)

if it DONT change then the rate of change (the 1st derivative) is constant and the basic equation is a straight line

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• no, there needs to be a change from positive negative or from negative to positive. the actual point where this change occurs (f''(x) = 0) is the point of inflection

a linear function's second derivative is 0 for all real numbers x but never changes signs, so it has no poi.

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• YES!!!

Also when first derivative is equated to zero, it means the curve is at a max or min.

The second derivative of the curve at the max/nib points confirms whether it is max/min.

For a maximum point the 2nd derivative is negative, and the minimum point is positive.

Taking y = x^2

dy/dx = 2x = 0

2x = 0

x = 0 , but is it a max/or min.

d2y /dx2 = (+)2 hence it is a minimum point.

• TomV
Lv 7
1 month agoReport

The 2nd derivative must change sign at the point for the point to be a point of inflection. y" = 0 is a necessary condition, but it is not sufficient.

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• Not necessarily. Suppose f(x) = 3x. Then f'(x) = 3

and f''(x) = 0. Clearly, the line defined by f(x)=3x

has no point of inflection.

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• The second derivative represents the rate of change of the slope.  If the second derivative is 0 at a specific point, the rate of change could be changing from + to - (or vice-versa).i.e. an inflection point or it could mean the rate of change of the slope is always 0...slope is constant ...straight line.

• TomV
Lv 7
1 month agoReport

It could also mean that the 2nd derivative is a parabolic function that is tangent to x-axis. Goes from positive (or negative) to zero and then goes from zero to positive (or negative).

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• No, not necessarily.

For instance, the second derivative of f(x) = mx+b is zero, but no point of inflection.

Or f(x) = x^4 * cos(x). This is f ''(0) = 0 but is not a pt of inflection.

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• Yes.

If a number, say a, is input for x to make the second derivative of f(x) become zero, there is a point of inflection at x = a.

• rotchm
Lv 7
1 month agoReport

No, it doesn't mean that there is an inflection. Its deconcerting to see that many have thumbed you up on such a gross error.

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