How do you find the minimum and maximum values of a function?
Answer: following all the steps that have been mentioned in the explanation below.
Let us look at an example to understand it better.
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 of x and find the max or min value.
Maximum value = 5
Minimum value = 1