Optimization
Why is the primary equation in an optimization problem called “primary”?
Because it’s the equation we derive.
True or False?
Critical points are always extremas, but extremas are not necessarily critical points.
False
What is an inflection point?
Point at which a function changes concavity (from up to down or down to up).
answer is 0
TRUE OR FALSE:
The Mean Value Theorem (MVT) implies Rolle’s Theorem.
True
The product is 176 and the sum is minimum.
x= 13.266, y = 13.266
PIC 1
The entire domain of f(x)
R / [1]
PIC 4
X > 6
How do we find it generally?
Direct subsitution
True or false:
The Mean Value Theorem is so named because it concerns the average (or "mean") rate of change of a function on an interval.
True
A pair of non-negative numbers with a product 160.
The chosen numbers should minimize the sum of 1 times the second number and the first number.
x= 12.65, y= 12.65
PIC 2
The entire domain of h(x)
R
PIC 5
(-4, 2)
How do we find it in rational expressions?
Divide by highest power.
True or false:
If f has a critical point at x=1, then f has a local minimum or maximum at x=1.
False
We want to construct a cylindrical can with a bottom but no top that will have a volume of 30 cm3. Determine the dimensions of the can that will minimize the amount of material needed to construct the can.
r= 2.1216, h=2.1215
Determine absolute extrema of PIC 3
Absolute maximum: 22, at x=1
Absolute minimum: -50, at x=-2
PIC 6
X = 3
true or false
If limx→∞f(x)=∞ then limx→∞sin(f(x)) does not exist.
False, exists and has value of 1
Check that function g(x) = cos(x) on the interval [- π/2 , 3π/2] satisfies all conditions of Rolle's theorem and then find all values x = c such that g '(c) = 0.
c1 = 0 and c2 = π