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Need help, Determine the points of inflection on the curve of the function
y = x^{3}
Consider the subsequent IVP. y' = f(t,y) , y(t 0 ) = y 0 If f(t,y) and ∂f/∂y are continuous functions in several rectangle a o - h o + h which is included in a
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The only possible inflexion points will happen where
(d^{2}y)/( dx^{2}) = 0
From specified function as:
(dy)/(dx) = 3x^{2} and (d^{2}y)/( dx^{2}) = 6x
Equating the second derivative to zero, we include
6x = 0 or x = 0
We test whether the point at that x = 0 is an inflexion point as follows
While x is slightly less than 0, ((d^{2}y)/(dx^{2})) < 0; it means a downward concavity
While x is slightly larger than 0, ((d^{2}y)/(dx^{2})) > 0; it means an upward concavity
Hence we have a point of inflexion at point x = 0 since the concavity of the curve changes as we pass from the left to the right of x = 0
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