**First Derivative:** For any function f (x) which is differentiable, the derivative f ' (x) is the slope of the tangent to the curve at any arbitrary point (x, y).

At a particular point (a, b) where b = f (a). The derivative f ' (a) will give us the slope of the tangent at the given point (a, b).

The slope of the tangent f ' (a) is positive if the tangent makes acute angle with the x-axis.

In the above graph Tangent AB makes acute angle with the x-axis, and hence the slope is positive. (i. e) f ' (x) > 0

The tangent CD makes obtuse angle with the x-axis, and hence the slope is negative. (i. e) f ' (x) < 0

**First Derivative Test for Local Extrema: **Let f be a differentiable function defined on an interval I and let a $\epsilon$ I, Then,

**(a)** x = a is a point of local maximum value of f, if

(i) f ' (a) = 0 and,

(ii) f ' (x) changes sign from positive to negative as x increases through a.

(i. e) 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.

**(b)** x = a is a point of local minimum value of f, if

(i) f ' (a) = 0 and,

(ii) f ' (x) changes sign from negative to positive as x increases through a.

(i. e) 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.

**(c)** If f ' (a) = 0 and f ' (x) does not change sign as x increases through a, that is f ' (x) has the same sign in the complete neighborhood of a, then a is neither a point of local maximum value nor a point of local minimum value. In fact, such a point is called a

**point of inflection**.

### First Derivative Test for Local Extrema Algorithm

**Step 1:** Put y = f (x)

**Step 2:** Find

$\frac{dy}{dx}$ = f ' (x)

**Step 3:** Equate

$\frac{dy}{dx}$ = f ' (x) = 0 and solve the equation for real values of x.

The real values of x are called the critical points of the function f (x).

Let c

_{1} , c

_{2}, c

_{3}, ..... be the critical points of the function f (x).

**Step 4:** If f ' (x) changes its sign from positive to negative as x increases through c

_{1} , then the function attains a local maximum at x = c

_{1} .

If f ' (x) changes its sign from negative to positive as x increases through c

_{1} , then the function attains a local minimum at x = c

_{1} .

If f ' (x) does not change sign as x increases throgh c

_{1} , then x = c

_{1} is neither a point of local maximum nor a point of local minimum.

In this case x = c

_{1} is called the point of inflection.

Similarly we can deal with other values of x.