Calculate a value of x that would give the minimum value for...

Calculate a value of $x$ that would give the minimum value for $S$ in the following expression: $$ \mathrm{S}=50\left(9.522 \times 10^{-6}\right)-\frac{50}{(50 \mathrm{X}+1)}\left(9.522 \times 10^{-6}\right)-\frac{60(0.22) \mathrm{X}}{30(24)} $$ Show that the function is a minimum or maximum.

Calculate a value of $x$ that would give the minimum value for $S$ in the following expression:
$$
\mathrm{S}=50\left(9.522 \times 10^{-6}\right)-\frac{50}{(50 \mathrm{X}+1)}\left(9.522 \times 10^{-6}\right)-\frac{60(0.22) \mathrm{X}}{30(24)}
$$
Show that the function is a minimum or maximum.

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves simplifying the given function into a more tractable form before applying calculus techniques. This step often includes combining like terms, factoring constants, or rewriting complex rational expressions so that differentiation and further analysis can be performed more easily.

Calculus Optimization

Calculus optimization is the process of finding the maximum or minimum values of a function by analyzing its derivatives. The strategy involves differentiating the function with respect to the variable of interest, setting the derivative equal to zero to find critical points, and then using further tests to determine if these points yield maxima or minima.

Critical Point Analysis

Critical points are the values of the variable at which the first derivative of the function is zero or undefined. These points are potential candidates for local minima or maxima. In optimization problems, identifying critical points is the first step in determining where the function reaches its extreme values.

Second Derivative Test

The second derivative test is a method used to classify the nature of a critical point. Once a critical point is found, the second derivative of the function is evaluated at that point. If the second derivative is positive, the function has a local minimum at the critical point; if it is negative, the function has a local maximum; and if it is zero, further investigation is needed.

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