## Description

1. (a) Using any software package plot the contours of f(x, y) = 25x

2 + 12y

2 − 20xy + 120y in

the range of (x,y) from (-15,-15) to (10,10).

(b) If there is a feasible region of x+y ≥ 0, what is the minima? (Note: On the hardcopy of

the contour plot draw the feasible region and find where the contour touches the feasible

region approximately, submit only the scanned copy of the annotated contour plot as

answer)

2. Find the maxima and minima of the following problem manually on a Graph Paper,

Objective function: f = y + 0.58x

Constraints:

(a) y − 2x − 5 ≤ 0

(b) y − x − 7.5 ≤ 0

(c) y + 5.5x − 66 ≤ 0

(d) (x − 7)2 + (y − 15)2 − 9 ≥ 0

(e) y ≥ 0

1

3. Express the function

f(x1, x2, x3) = −x

2

1 − x

2

2 + 2x1x2 − x

2

3 + 6x1x3 + 4×1 − 5×3 + 2

in matrix form as

f(X) = 1

2

XT

[A]X + B

T X + C

and determine whether the matrix A is positive definite, negative definite, or indefinite.

4. The potential energy of a particle moving along the x direction is given by,

U(x) = 3x

2 − x

3

Plot the potential energy as a function of x. Identify all the possible equilibrium points, and

label the stable equilibrium position.

5. Find the second-order Taylor’s series approximation of the function

f(x1, x2, x3) = x

2

2×3 + x1e

x3

at the point (1,0,-2).

6. For a triangle ABC, find the maximum value of sin(A) + sin(B) + sin(C). Formulate it as a

constrained optimization problem.

7. Minimize

f(X) = 1

2

(x

2

1 + x

2

2 + x

2

3

)

subject to

g1(X) = x1 − x2 = 0,

g2(X) = x1 + x2 + x3 − 1 = 0

by

(a) direct substitution

(b) constrained variation

(c) Lagrange multiplier method

8. (a) Minimise the function

f(x, y) = x

2 + y

2

subject to

g(x, y) = xy = 1,

using the Lagrange multiplier method. Find the solution point(s) and the corresponding

Lagrange multiplier(s).

(b) Find ∇f and ∇g at the solution point. How are they related? What implication does

this have on the contour lines of f and g ?

(c) How does the relation between the gradients ∇f and ∇g computed at the solution point

compare with the Lagrange multiplier ?

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