One Variable Optimisation

Question 1

When is a function convex?

Answer 1

If any two x₁ and x₂ and any t[0,1]
tf(x₁) + (1 - t)f(x₂) ≥ f(tx₁ + (1 - t)x₂)

Question 2

For the interior points of a differentiable function, what are the necessary conditions for a point, c, to be a maximum?

Answer 2

f(c) must be a stationary point.

Question 3

What the only necessary condition to guarantee the existence of an extreme point?

Answer 3

Continuity.

Question 4

If a function is concave, where will the function lie between any two points joined on a line?

Answer 4

If the line is below or on the graph.

Question 5

If a function is convex, where will the line segement that joins two points on the graph be?

Answer 5

Above or on the graph.

Question 6

If a function is twice continuously differentiable, how can f(x) be proved to be convex (concave) over D?

Answer 6

f’‘(x) ≥ 0 for all x ∈ D.

(f’‘(x) ≤ 0 for all x ∈ D.)

Question 7

How is f(c) a maximum (minimum) point of f(x)?

Answer 7

If f(x) ≤ f (c).

(If f(x) ≥ f(c)).

Question 8

How to define a stationary point, c, in a function f(x)?

Answer 8

f’(c) = 0.

Question 9

Suppose f(x) is differentiable over an interval, I, and that c is an interior point of I. How can c be an extreme point in I?

Answer 9

If f’(c) = 0

This is known as the first-order condition (FOC)

Question 10

What is a necessary condition for an interior point, c, of a differentiable function, f(x), to be an extreme point?

Answer 10

It must also be a stationary point.

(f’(c) = 0)

Question 11

Where do the FOC’s fail?

Answer 11

• If f(x) is not differentiable over the whole domain
• If f(x) has stationary points such as local max/min or inflection points

Question 12

Why do FOC’s fail when f(x) is not differentiable over the whole domain?

Answer 12

Let f(x) is defined over [a,b] but not differentiable at d.

Let f(d) be the minimum of f(x), but it is not a stationary point as f(x) is not differentiable at f(d).

Therefore, the FOC’s fail.

Question 13

Why do FOC’s fail when f(x) has stationary points such as local max/min or inflection points?

Answer 13

Let f(x) have S stationary points: x₀, x₁, x₂, …, x_S, but none are extreme points over the domain [a,b].

The minimum of f(x) is a, and maximum is b. (i.e. not interior points).

If f(x₀) is a local min., f(x₁) is a local max. etc.

Therefore, f’(x) = 0 is not a sufficient condition to identify global extreme points.

Question 14

What is the extreme value theorem?

Answer 14

If f is a continuous function over a closed bounded interval [x₀, x₁], then there exists a point c ∈ [x₀, x₁] where f has a minimum and a point d ∈ [x₀, x₁] where f has a maximum so that:

f(c) ≤ f(x) ≤ f(d) for all x ∈ [x₀, x₁].

Question 15

When do you use the extreme value theorem?

Answer 15

When interested if an extreme points exists, rather than it’s actual location.

Question 16

Why would you use the extreme value theorem?

Answer 16

When the function is not explicit or it’s derivative cannot be computed in closed form.

Question 17

What is the mean value theorem?

Answer 17

If f is continuous over closed bounded interval [ [x₀, x₁] and differentiable in the open interval (x₀, x₁), then there exists at least one interior point, c ∈ (x₀, x₁) such that:

f’(c) = (f(x₁) - f(₀))/ (x₁ - x₀)

Question 18

What happens if f(x₀) = f(x₁)?

(And the function is continuous and differentiable over some interval [x₀, x₁])

Answer 18

At least one stationary point exists.

(Rolle’s theorem)

Question 19

Is it necessary/sufficient for an extreme point to be a local extreme point?

Answer 19

Necessary and sufficient.

Question 20

How to define an inflection point?

Answer 20

If, in the function, f, there exists an interval (a,b) around c such that the function is convex/concave on (a,c) and concave/convex on (c,b)

Question 21

What are the sufficient conditions for the FOCs to guarantee or maximum/minimum?

Answer 21

If f(x) is convex/concave over an interval, I.
If c is a stationary point for f in the interior of I, then c is a minimum (maximum) point for f in I.

Check if f is convex/concave over the interval, then use FOCs.

Question 22

If f is differentiable over I and also satisfies this:

If f(x) is convex/concave over an interval, I.
If c is a stationary point for f in the interior of I, then c is a minimum (maximum) point for f in I.

Is the FOC necessary/sufficient?

Answer 22

The FOC is necessary and sufficient.

Question 22

If f satisfies the constraints on curvature over I such that f(x) is convex/concave over an interval, I.
If c is a stationary point for f in the interior of I, then c is a minimum (maximum) point for f in I, is the FOC necessary/sufficient?

Answer 22

Sufficient, but not necessary.

The function need not be differentiable at an optimum.

f(x) maximum/minimum might be at the bound of the function.

Question 23

If f(x) has a stationary point at x = c, f’(c) = 0, how can you assign c to local max / local min / inflection point?

Answer 23

f^j(x) where j is the j^th derivative and n the smallest number such that fⁿ(c) ≠ 0.

c is:
* local max. if n is even and fⁿ(c) < 0
* local min. if n is even and fⁿ(c) > 0
* inflection point is n is odd

Question 24

What are the slope constraints?

Answer 24

If f(x) is differentiable over I and c is an interior point of I. * If f'(x)≥ 0 for all x ∈ I where x ≤ c, and f′(x) ≤ 0 for all x ∈ I where x ≥ c, then c is a maximum point in I. * If If f′(x) ≤ 0 for all x ∈ I where x ≤ c and f'(x) ≥ 0 for all x ∈ I where x ≥ c, then c is a minimum point in I. ## Footnote If f increasing when x ≤ c, and f decreasing when x ≥ c, then c is max and vice-versa.

Question 25

Are slope constraints necessary/sufficient for finding extreme points?

Answer 25

Sufficient.