How to Calculate Maximum & Minimum
The maximum and minimum of any function f(X) correspond to points (one or several values of the variable X) where the first derivative of the function (denoted as f'(X)) turns to zero. These points are called function extremums. The second derivative (denoted as f''(X)) should be also computed to determine if a particular extremum is maximum or minimum. For example, calculate the minimum and maximum of the function f(X) = X^3-4X^2-3X.
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Instructions
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1
Calculate the first derivative of the function f(X). Consult the "Derivative" resources below to find a differentiation formula corresponding to your function. Alternatively, you can calculate derivatives using the online calculator (see Section 2).
In our example, the appropriate differentiation formula is d(CX^p)/dX = pCX^(p-1). C is any constant number.
F'(X) = 3 x X^2-4 x 2X-3 = 3X^2-8X-3. -
2
Solve the equation f'(X) = 0. Note that the solution procedure would depend on a particular equation. The number of solutions of this equation is equal to the number of extremums for the function f(X).
In our example, it is a quadratic equation: 3X^2-8X-3 = 0. Generally, it has two solutions (denoted as X1 and X2) defined as:
X1 = [-(-8)+sqrt(8^2-4x3x(-3) ]2x3 = [8+sqrt(64+36)]/6 = 18/6 = 3.
X1 = [-(-8)-sqrt(8^2-4x3x(-3) ]2x3 = [8-sqrt(64+36)]/6 = -2/6 = -1/3.
("Sqrt" is an abbreviation for the root square math operation.) -
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3
Calculate the second derivative of the function f(X) by the differentiation of the first derivative function (obtained in Step 1). Use the same approaches as in Step 1.
In our example, the second derivative would be:
F''(X) = 3x2X-8-0 = 6X-8. -
4
Calculate the values of the second derivative function at the points of the extremums. If this function is less than zero, the extremum is the maximum. If it is greater than zero, the extremum is the minimum.
In our example,
F''(X1) = 6x3-8 = 10. 10 is greater than 0, hence X1 = 3 is the minimum.
F''(X2) = 6x(-1/3)-8 = -10. -10 is less than 0, hence X2 = -1/3 is the maximum. -
5
Calculate the maximum and minimum values of the function f(X) at "X" identified in Step 4.
In our example,
The function maximum (at X = -1/3) = (-1/3)^3-4(-1/3)^2-3(-1/3) = -1/27-4/9+1 = 14/27.
The function minimum (at X = 3) = 3^3-4(3^2)-3x3 = 27-36-9 = -18.
How to use a derivative calculator
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Navigate to the derivative calculator using the link in Resources.
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Type your function in the field under "Enter a function to differentiate."
Note: You must use the small "x" to denote a variable. Accordingly, the multiplication sign "x" must not be used. The full list of permitted operators is given on the same Web page.
In our example, enter the function like this: "x^3-4x^2-3x" -
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Click "Go!" and read the derivative on the next screen. In our example, you should get: f'(x) = 3*x^2-8*x-3
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