The special case of x⁴
Key Questions

The stationary point is
#(0,0)# and it is a minimum point.The first task in finding stationary points is to find the critical points, that is, where
#f'(x)=0# or#f'(x)# DNE:#f'(x)=4x^3# using the power rule
#4x^3=0#
#x=0# is the only solutionThere are 2 ways to test for a stationary point, the First Derivative Test and the Second Derivative Test.
The First Derivative Test checks for a sign change in the first derivative: on the left the derivative is negative and on the right the derivative is positive, so this critical point is a minimum.
The Second Derivative Test checks for the sign of the second derivative:
#f''(x)=12x^2#
#f''(0)=0# There is no sign, so the second derivative doesn't tell us anything in this case.
Since there are no other minimums and
#lim_(x>oo)f(x)=lim_(x>oo)f(x)=oo# . The stationary point is an absolute minimum.
Questions
Graphing with the Second Derivative

Relationship between First and Second Derivatives of a Function

Analyzing Concavity of a Function

Notation for the Second Derivative

Determining Points of Inflection for a Function

First Derivative Test vs Second Derivative Test for Local Extrema

The special case of x⁴

Critical Points of Inflection

Application of the Second Derivative (Acceleration)

Examples of Curve Sketching