The potential energy of a system of two particles is given by u(x)=a/x2-b/x. Find minimum potential energy of the system, where x is the distance of separation and a, b are positive constants.

I think you probably mean:

U = a/x^2 -b/x

dU/dx = -2ax/x^4 +b/x^2
where is that zero?

-2a/x^3 + b x/x^3 = 0
for zero x=2a/b
U = a/(4a^2/b^2)- b/(2a/b)
= a/(4a^2/b^2) - 2a/(4a^2/b^2)
= -a/ (4 a^2/b^2)
= -b^2/4a
or
U -a/x^2

To find the minimum potential energy of the system, we need to find the value of x that minimizes the potential energy function u(x) = a/x^2 - b/x.

To begin, let's differentiate u(x) with respect to x:
du/dx = -2a/x^3 + b/x^2

To find the minimum potential energy, we need to find the value of x where du/dx = 0. Let's set du/dx equal to zero and solve for x:
-2a/x^3 + b/x^2 = 0

Multiplying through by x^3, we get:
-2a + bx = 0

Now, let's solve for x:
bx = 2a
x = 2a/b

Therefore, the distance of separation x that minimizes the potential energy is x = 2a/b.

To find the minimum potential energy, we substitute this value back into the potential energy function u(x):
u(min) = a/(2a/b)^2 - b/(2a/b)
= a/(4a^2/b^2) - b/2a
= (b^2)/(4a) - b/2a
= b^2/(4a) - 2b/(4a)
= (b^2 - 2b)/(4a)

Therefore, the minimum potential energy of the system is (b^2 - 2b)/(4a).

To find the minimum potential energy of the system, we need to find the value of x that minimizes the potential energy function u(x).

Step 1: Find the derivative of the potential energy function u(x) with respect to x.

u'(x) = d/dx [a/x^2 - b/x]

To find the derivative, we need to use the quotient rule. The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative of f(x) is given by:

f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]^2.

In our case, g(x) = a and h(x) = x^2 - b/x. Taking the derivatives, we have:

g'(x) = 0 (since a is a constant)
h'(x) = 2x + b/x^2

Applying the quotient rule, we get:

u'(x) = [0 * (x^2 - b/x) - a * (2x + b/x^2)] / (x^2 - b/x)^2

Simplifying further, we get:

u'(x) = -2ax - ab/x^2 / (x^2 - b/x)^2

Step 2: Find the critical points of u(x) by setting u'(x) = 0 and solve for x.

-2ax - ab/x^2 = 0

Multiply through by -x^2 to get rid of the fraction:

2ax^3 + ab = 0

Divide through by a:

2x^3 + b/a = 0

Solving this equation for x is not straightforward unless you are given a specific value for b/a.

Step 3: Determine whether the critical point(s) found is/are minimum or maximum by analyzing the second derivative of u(x).

Take the derivative of u'(x) with respect to x:

u''(x) = d/dx [-2ax - ab/x^2] = -2a + 2ab/x^3

Evaluate u''(x) at the critical point(s) found in step 2.

If u''(x) is positive, then the critical point is a minimum. If u''(x) is negative, then the critical point is a maximum. If u''(x) is zero, further analysis is required.

Without knowing the specific values of a and b, we cannot determine the exact critical point(s) and whether it is a minimum or maximum.

Thanks