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FaroukYasser
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I was wandering, can $$\ \frac { d^{ 2 }y }{ dx^{ 2 } } | _{ x={ x }_{ 0 } }=\quad 0$$ if $$\frac { dy }{ dx }| _{ x={ x }_{ 0 } }\neq \quad 0$$ and if so, what does this translate to geometrically?
Samy_A said:##y=x-x_0##
This means that the slope of the curve is not changing at that pointFaroukYasser said:I was wandering, can $$\ \frac { d^{ 2 }y }{ dx^{ 2 } } | _{ x={ x }_{ 0 } }=\quad 0$$
This means the slope is not zero at the point. So @Samy_A 's example of a sloped straight line satisfies both conditions at every point.if $$\frac { dy }{ dx }| _{ x={ x }_{ 0 } }\neq \quad 0$$ and if so, what does this translate to geometrically?
Nothing. That is exactly what it means. sin(x) has an inflection point at every x= n * pi. Those are all inflection points with non-zero slopes.FaroukYasser said:In other words, what does f''(x_0) = 0 tell us about that point other than it being a likely inflection point.
The second derivative geometrically identifies concavity. f'' > 0 or f''<0 on an interval identifies concave up or down, respectively. In a case where f'' is continuous, the only way for concavity to switch is for f'' to pass through zero. So f'' = 0 at a point might be a case where f'' is changing sign, indicating an inflection point (change of concavity). Of course, f'' might not change sign nearby so it might not be an inflection point. I wouldn't say such a point is a "likely" inflection point but a "possible" inflection point.FaroukYasser said:Thanks. But I was hoping for a non linear function. Or in other words, I am wandering what this means geometrically. I know that if a point is an inflection point then its second derivative is 0 but the converse doesn't necessarily hold. In other words, what does f''(x_0) = 0 tell us about that point other than it being a likely inflection point.
Thanks :)
Yes, it is possible for f''(x_0) to be equal to 0 even if f'(x_0) is not equal to 0. This occurs when the slope of the tangent line at x_0 is changing direction, causing the second derivative to be 0 at that point.
The value of f'(x_0) does not directly affect the value of f''(x_0). However, the behavior of f'(x) near x_0 can provide information about the value of f''(x_0), such as whether it is positive or negative.
Yes, it is possible for f'(x_0) and f''(x_0) to have the same value. This occurs when f(x) is a constant function, as both the first and second derivatives will be 0 at any point x_0.
If f''(x_0) = 0 and f'(x_0) is not equal to 0, it means that the slope of the tangent line at x_0 is changing from positive to negative or vice versa. This indicates a point of inflection on the graph of f(x), where the concavity changes.
The values of f'(x_0) and f''(x_0) provide information about the behavior of f(x) at x_0. If both derivatives are positive, it suggests that f(x) is increasing and concave up at x_0. If f'(x_0) is positive and f''(x_0) is negative, it suggests that f(x) is still increasing but the concavity is changing from up to down. Similarly, if both derivatives are negative, it suggests that f(x) is decreasing and concave down at x_0. If f'(x_0) is negative and f''(x_0) is positive, it suggests that f(x) is still decreasing but the concavity is changing from down to up.