Find the minimum of the function f(x)= x^2 – ((54 ) / (x) ).
To find the minimum of the function f(x) = x^2 - (54 / x), you can follow these steps:
1. Start by taking the derivative of the function f(x) with respect to x. This will give us the slope of the function at any given point.
f'(x) = 2x + 54 / x^2
2. Set the derivative equal to zero and solve for x to find the critical points. In this case, we have:
2x + 54 / x^2 = 0
Multiply both sides by x^2 to eliminate the fraction:
2x^3 + 54 = 0
Subtract 54 from both sides:
2x^3 = -54
Divide both sides by 2:
x^3 = -27
Take the cube root of both sides:
x = -3
(Note: We have found one critical point x = -3, but there could be more critical points. We will verify this later.)
3. Use the second derivative test to determine whether the critical point x = -3 is a minimum or maximum. Take the derivative of the derivative (also known as the second derivative):
f''(x) = 2 - 108 / x^3
Substitute the critical point x = -3 into the second derivative:
f''(-3) = 2 - 108 / (-3)^3
= 2 - 108 / -27
= 2 - (-4)
= 6
Since the second derivative is positive at x = -3, it means that the critical point is a local minimum.
4. To confirm that x = -3 is the absolute minimum, we need to check the behavior of the function as x approaches positive or negative infinity.
As x approaches positive infinity, the term x^2 becomes dominant, and therefore, the function approaches infinity.
As x approaches negative infinity, the term -(54 / x) becomes dominant, and the function approaches negative infinity.
Since the function tends to negative infinity as x approaches negative infinity, and x = -3 is a local minimum, it is the absolute minimum.
Therefore, the minimum value of the function f(x) = x^2 - (54 / x) is f(-3) = (-3)^2 - (54 / -3) = 9 + 18 = 27.