Find the stationary points of function E(a,b) = a^4 - 2a^2 + 2b^2
and classify each of these as either a minimum, maximum or saddle point.
To find the stationary points of a function, you need to find the values of a and b where the partial derivatives of the function with respect to each variable are equal to zero.
The partial derivative with respect to a (denoted as ∂E/∂a) is found by differentiating the function E(a, b) with respect to a and treating b as a constant. Similarly, the partial derivative with respect to b (∂E/∂b) is found by differentiating the function E(a, b) with respect to b and treating a as a constant.
Let's find the partial derivatives first:
∂E/∂a = 4a^3 - 4a
∂E/∂b = 4b
Now, set these equations equal to zero and solve for a and b:
∂E/∂a = 4a^3 - 4a = 0
∂E/∂b = 4b = 0
For ∂E/∂b = 4b = 0, the only solution is b = 0.
Now, for ∂E/∂a = 4a^3 - 4a = 0, we can factor out 4a:
4a(a^2 - 1) = 0
This gives us two possible solutions:
1. a = 0
2. a^2 - 1 = 0 => a^2 = 1 => a = ±1
So, the stationary points of the function E(a, b) are (0, 0), (1, 0), and (-1, 0).
To classify the nature of each stationary point, we can use the second partial derivative test. This involves finding the second partial derivatives ∂^2E/∂a^2, ∂^2E/∂a∂b, and ∂^2E/∂b^2, and evaluating them at the stationary points.
∂^2E/∂a^2 = 12a^2 - 4
∂^2E/∂a∂b = 0 (since ∂E/∂b is constant)
∂^2E/∂b^2 = 4
Evaluating these second partial derivatives at each of the stationary points:
- At (0,0):
∂^2E/∂a^2 = 12(0)^2 - 4 = -4
∂^2E/∂a∂b = 0
∂^2E/∂b^2 = 4
Since the second partial derivative test requires evaluating a determinant (Hessian matrix), and the determinant is negative for the point (0, 0), the point (0, 0) is classified as a saddle point.
- At (1,0):
∂^2E/∂a^2 = 12(1)^2 - 4 = 8
∂^2E/∂a∂b = 0
∂^2E/∂b^2 = 4
Since the second partial derivative test requires evaluating a determinant (Hessian matrix), and the determinant is positive for the point (1, 0), and ∂^2E/∂a^2 > 0, the point (1, 0) is classified as a minimum.
- At (-1,0):
∂^2E/∂a^2 = 12(-1)^2 - 4 = 8
∂^2E/∂a∂b = 0
∂^2E/∂b^2 = 4
Since the second partial derivative test requires evaluating a determinant (Hessian matrix), and the determinant is positive for the point (-1, 0), and ∂^2E/∂a^2 > 0, the point (-1, 0) is also classified as a minimum.
So, the stationary points of the function E(a, b) = a^4 - 2a^2 + 2b^2 are:
- (0, 0) as a saddle point
- (1, 0) and (-1, 0) as minimum points.