determine the nature of the stationary point in the curve y=x^4 using first derivative test?

Question

a) determine the nature of the stationary point in the curve y=x^4 using first derivative test

b) what happens if you use second derivative test to determine nature of the stationary point.

i know that first derivative is specifically used to find gradient and second for concavity so whats the reason if any to use second derivative??

cidyah · Answer

y=x^4
y' = 4x^3 = 0
x=0 is the stationary point

As x passes through the neighborhood of 0, y' = 4x^3 changes from negative to positive, so f(x) has a global minimum at x=0

y'' = 12x^2
when x=0 , y'' =0
We cannot use the second derivative test whether we have a minimum or maximum

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