How do you find the minimum and maximum values of a function?

Solution:

A function means a correspondence from one value x of the first set to another value y of the second set. The highest value of a function is considered the maximum value of the function, and the lowest value of the function is considered the minimum value of the function.

There are some techniques for determining a function's maximum or minimum value.

• Find the derivative of the function and equip it to zero.
• Find the roots of the differentiated equation.
• Double differentiate the original function and replace the root values ​​in the second differentiated expression.
• If the value turns out to be negative, the particular root value Maximum occurs. Then replace the value in the original expression to get the maximum of the function.
• If the value of the double derivative after root substitution is positive, the minimum occurs. Then replace the value of the original equation to get the minimum value of the function.
• If the second derivative is zero: then find the highest derivatives of the function and replace the value of the root in the expression for the nth order derivative. If it is positive, it will give the maximum function to the particular root.

Example: y = x³ - 3x² + 5

y = x³ - 3x² + 5 -------------(1)

Differentiate both of side (i), w.r.t - x.

⇒ dy/dx =  d/dx (x³) - d/dx (3x²) + d/dx (5)

⇒ 3x² - 6x + 0

⇒ dy/dx = 3x² - 6x ----------------(2)

dy/dx = 0 given, critical points.

⇒ 3x² - 6x = 0

⇒ 3x (x - 2) = 0

⇒ x = 0,2

The critical points is 0 & 2.

Differentiate both of side (iI), w.r.t - x.

⇒ d²y/dx² = d/dx (3x²) - d/dx (6x)

⇒ d²y/dx² = 6x - 6

Now, put the value of x in (1) and find the max or min value.

Maximum value = 5

Minimum value = 1

How do you find the minimum and maximum values of a function?

Summary:

To find the maximum and minimum values of a function, follow these steps in order: Find the first derivative of the function, find the roots of the differentiated function, which form the critical point. Find the second derivative of the given function, apply the critical obtained in the second derivative of the function. Find the maximum/minimum value by substituting the critical points in the original function if the critical point substituted in the second derivative is positive or negative accordingly.