Critical Points
What defines a critical point?
f'(x)=0 or undefined
f''>0 means
Concave Up
Goal
Max or min
Differentiate wrt
MVT says
Instantaneous=average
Critical point of x^2-4x
x=2
f''<0 means
Concave Down
First step
Define variables
Sphere volume formula
4/3πr^3
Formula
(f(b)-f(a))/(b-a)
Derivative sign test tells
Increasing/decreasing
Inflection point means
Concavity changes
Solve by
Set f'=0
dV/dt formula
4πr^2 dr/dt
Requires
Continuous + differentiable
At max/min derivative is
0
Second derivative used for
Concavity
Check also
Endpoints
Key idea
Variables related
f(x)=x^2 [1,3] c
2
Critical point meaning
Possible extrema
x^3 inflection point
x=0
Rectangle max area shape
Square
Given dr/dt find
Another rate
Meaning
Slope match exists