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Fundamentals of Stress and Vibration
[A Practical guide for aspiring Designers / Analysts] 1. Mathematics for Structural mechanics
Taking ‘x’ inside the root we get:

3 3
2 2 2 2 8 2
3
3 = xy = x 3 d − x 3
2
xy = x 3 d − x 2+ 3 2

2 8

2
Therefore the term under the root is: x 3 d − x 3

Differentiating with respect to x we get:

3
d y x 8 5 2 1
−
2
= − ∗ x 3 + d ∗ ∗ x 3 = 0
dx 3 3

d
Therefore, we get x =
2
2
Substituting the value of ‘x’ in the equation x + y = d , we get:
2
2
d
y = 3
2

The optimum or maximum value of xy is given by:
3

d d 3 d 4
3
xy = ∗ 3 = 3 3
2 2 16

Example 2: Find the minimum value of f x, y = x + y + 2xy , using gradient approach.
2
2
2

Note: Though this is a simple case, the entire procedure of using the Hessian matrix is well
demonstrated.

Solution 2: in order to solve this situation we would use the Hessian matrix, which is given as
follows:

2
2
∂ f ∂ f
∂x 2 ∂x ∂y
2
Hessian Matrix = ∂ f ∂ f
2
2
∂y ∂x ∂y
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QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,
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