Application of Derivatives - II

• We can apply derivatives to determine approximate value of certain quantities and to find maximum and minimum values of a function in an interval.
• The differential of y is denoted by dy and is defined by dy = f ′(x) dx or dy = f ′(x) ∆x, where dy approximately equal to ∆y.
• First Derivative Test for Maxima and Minima:

Let f (x) be a function defined on an open interval I and continuous at a critical point ‘a’ in I. Then

• If f ′(x) changes sign from positive to negative as x increases through a, i.e.,

if f ′(x) > 0 at every point sufficiently close to and to the left of a, and f ′(x) < 0 at every point sufficiently close to and to the right of a, then a is a point of local maxima.

• If f′(x) changes sign from negative to positive as x increases through a, i.e., if f′(x) < 0 at every point sufficiently close to and to the left of a, and f ′(x) > 0 at every point sufficiently close to and to the right of a, then a is a point of local minima.
• Second Derivative Test  For Maxima and Minima:
• Let f(x) be a function defined on an interval I and a Î I. Let f(x) be twice differentiable at a. Then
• x = a is a point of local maxima, and if f ′(a) = 0 and f ″(a) < 0, then f(a) is the local maximum value of f (x).
• x = a is a point of local minima, and if f′(a) = 0 and f ″(a) > 0, the f (a) is the local minimum value of f (x).
• The test fails if f'(a) = 0 and f" (a) = 0.

       Keywords: Maximum and minimum values of a function, First derivative test, Second derivative test