Rate of Change of Quantities

 

Increasing and Decreasing Functions

Tangents

Normals

Maxima and Minima

First Derivative Test

Let f be a function defined on an open interval I. Let f be continuous at a critical point cin I. Then

If f′(x) changes sign from positive to negative as x increases through c, i.e., if f′(x) > 0 at every point sufficiently close to and to the left of c, and f′(x) < 0 at every point sufficiently close to and to the right of c, then c is a point of local maxima.

If f′(x) changes sign from negative to positive as x increases through c, i.e., if f′(x) < 0 at every point sufficiently close to and to the left of c, and f′(x) > 0 at every point sufficiently close to and to the right of c, then c is a point of local minima.

If f′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. Infact, such a point is called point of inflexion.

Second Derivative Test

Let f be a function defined on an interval I and c ∈ I. Let f be twice differentiable at c. Then

x= c is a point of local maxima if f′(c) = 0 and f″(c) < 0. The values f(c) is local maximum value of f.

x= c is a point of local minima if f′(c) = 0 and f″(c) > 0. In this case, f(c) is local minimum value of f.

The test fails if f′(c) = 0 and f″(c) = 0. In this case, we go back to the first derivative test and find whether c is a point of maxima, minima or a point of inflexion.