Maximum and Minimum Values of A Function Physics Notes

Maximum and Minimum Values of a function:
A function f(x) is said to have a relative maximum value at x = a, if f(a) is greater than any value immediately preceding or following.

Maximum and Minimum Points

A function f(x) is said to have a relative minimum value at x= b, if f(b) is less than any value immediately preceding or following.

The tangent to the curve in figure is horizontal (see point A and B). The slope of each tangent line, i.e., the derivative when evaluated at A or Bis zero (0).
i. e., f'(x) = 0.
At points immediately to the left of a maximum, the slope of the tangent is positive:
f'(x) > 0
At points immediately to the right of a maximum, the slope of the tangent is negative:
f'(x) < 0
In other words, at a maximum, f’ (x) changes the sign from + to – .
At a minimum, f’ (x) changes the sign from – to +.

We observe that at a maximum, at A the graph is concave upward.

The value of x at which the function has either a maximum or a minimum is called a critical value. In the figure, the critical values are x = a and x = b.

The sufficient condition for extreme values of a function at a critical value a:

The function has a minimum value at x = a if f'(a) = 0 and f”(a) – a positive number
The function has a maximum value at x = a if f'(a) = 0 and f”(a) = a negative number.