Topic 1: One-Variable optimisation

Question 1

How do you know if a function is convex (2)

Answer 1

-If for any x1 and x2 and t e [0,1], tf(x1) + (1-t)f(x2) > f(tx1 + (1-t)x2) (the formula basically saying the averages of the 2 points > the function t (t is a weight) % in between the 2 points)
-if the line segment that joins any 2 points on the graph is above/on the graph

Question 2

What is the first order condition (2)

Answer 2

-for an interior point of a differentiable function, a necessary condition for an extreme point is that it is also a stationary point
-Not sufficient due to local max/mins and POI)

Question 3

What is the extreme value theorem (2)

Answer 3

-If f is continuous over a closed bounded interval, there exists a point where f has a minimum and x has a maximum
-This is sufficient but not necessary for extreme points to exist

Question 4

What is the mean value theorem + Rolle’s theorem (2)

Answer 4

if f is continuous over a closed bounded interval, and differentiable, there exists one interior point where the slope at that point = the average slope across the whole interval
-Therefore, if the 2 boundary points are equal, a stationary point must exist

Question 5

How can we assign stationary points to different types (3)

Answer 5

Let n be the smallest number where (differentiated n times) f^n (c) isn’t 0
-C is a local min if n is even and f^n (c) > 0
-C is a local max if n is even and f^n (c) < 0
-C is an inflection point if n is odd