Critical Points
A point where f'(x)=0 or f'(x) is undefined is called
What is a critical point
This occurs if f'(x) changes from positive to negative
What is a local maximum
What does f''(x) tell you
What is concavity
On a closed interval, where do absolute extrema occur
What are at critical points or endpoints
A calculus tool that is the most important in optimization
What are derivatives?
The critical points of f(x)=x2-4x+3.
What is x=2
What occurs if f'(x) changes from negative to positive
What is a local minimum
If f''(c) > 0, what happens at c
What is local minimum
Find the absolute maximum of f(x)=x² on [-1,2]
What is 4 at x=2
A rectangle has perimeter 20. What gives the maximum area
What is 5 x 5
The critical points of f(x)=x3-3x2+2
What is x=0 and x=2
Use the first derivative test to classify x=1 for f(x)=x3-3x
What is a Local minimum
If f''(c) < 0, what happens at c
What is local maximum
Find the absolute minimum of f(x)=x²-4x on [0,5]
What is -4 at x=2
What shape minimizes surface area for a cylinder with fixed volume
What is height equals diameter
The critical points of f(x)=x4-8x2+16
What is x=-2, 0, 2
The absolute extrema of f(x)=x3 on (-∞,∞)
What is no absolute extrema
If f''(c) = 0, what can you conclude
What is inconclusive
Why can a function on an open interval not have an absolute maximum
What is endpoints are not included
A farmer has 100 m of fencing for a rectangle against a barn. What dimensions maximize area?
What is the width = 25 m and length = 50 m
The critical points of f(x)=∛x(x-4)
What is x=0 and x=1
The absolute extrema of f(x)=x3-3x2+2 on [0,2]
What is the absolute max = 2 at x=0 and absolute min = -2 at x=2
The first step in the second derivative test
What is find critical points (where f'(x)=0)
The absolute extrema of f(x)=x³-3x on [-2,2]
What is the absolute max = 2 at x=-1; absolute min = -2 at x=1
A box has volume 32 and minimum surface area. What shape is it
What is a cube with side ∛32