Introduction:
In this problem, we are required to find the maximum value of the expression 1/x² 5x 10. To solve this problem, we can use the concept of differentiation to find the critical point of the function and then check for the maximum value.

Finding the Critical Point:
To find the critical point, we need to differentiate the given expression with respect to x and equate it to zero. So, let's differentiate the expression:
f(x) = 1/x² 5x 10

f'(x) = -2/x³ + 5

Now, equating f'(x) to zero, we get:
-2/x³ + 5 = 0
-2/x³ = -5
x³ = 2/5
x = (2/5)^(1/3)

Checking for Maximum Value:
To check whether the critical point is a maximum or not, we need to check the second derivative of the function at the critical point. If the second derivative is negative, then the critical point is a maximum.

So, let's differentiate f'(x) with respect to x:
f''(x) = 6/x^4

Now, substituting the critical point x = (2/5)^(1/3) in f''(x), we get:
f''((2/5)^(1/3)) = 6/[(2/5)^(4/3)]

Since (2/5)^(4/3) is a positive number, f''((2/5)^(1/3)) is also positive. Therefore, the critical point x = (2/5)^(1/3) is a minimum.

Conclusion:
Since the critical point is a minimum, it means that the maximum value of the function occurs at the endpoints of the domain. As x approaches positive or negative infinity, the function approaches zero. Therefore, the maximum value of the expression 1/x² 5x 10 is 10, which occurs at x = 0.

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