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Consider f(x) = 2x^{2} , ∀ x ∈ R

From the graph, we have f(x) = 0 if x = 0 and f(x) ≥ 0 ∀ x ∈ R

The minimum value of f is 0 and the point of minimum value of f is x = 0.

The graph of function f* *has no maximum value, and hence, no point of maximum value of f in R.

Let f be a function defined on an open interval D. Let f be continuous at a critical point, a, in D.

**Local minima**

If f '(x) changes its 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 the point of local minima.

The local minimum value is f(a).

**Local maxima**

If f '(x) changes its 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 the point of local maxima.

The local maximum value is f(a).

'0' is neither a point of local maxima nor a point of local minima.

'0' is the point of inflection.

**Point of inflection**

If f '(x) does not change its sign as x increases through critical point "a", then "a" is known as the point of inflection.

At point "a", f has neither a point of local maxima nor a point of local minima.

**Steps for finding local maxima and local minima:**

**Step1:** Find the derivative of f(x), i.e. f '(x).

**Step2:** Let a_{1},a_{2},a_{3},... ... ..., a_{n} be the roots of f '(x) = 0 . Here, a_{1},a_{2},a_{3},... ... ..., a_{n} are called the critical points or stationary points of x.

**Note:** The local maxima or local minima of f(x) can be attained at some or all these stationary points of x.

**Step 3:**Consider x = a_{1}. If f '(x) > 0 at the left neighbourhood values of a_{1} and if f '(x) < 0 at the right neighbourhood values of a_{1} , then x = a_{1} is the point of local maxima.

Consider x = a_{1}. If f '(x) < 0 at the left neighbourhood values of a_{1} and if f '(x) > 0 at the right neighbourhood values of a_{1}, then x = a_{1} is the point of local minima.

**Note:** If f '(x) does not change its sign as x increases through a_{1 }, then x = a_{1} is neither a point of local maxima nor a point of local minima.

Consider f(x) = 2x^{2} , ∀ x ∈ R

From the graph, we have f(x) = 0 if x = 0 and f(x) ≥ 0 ∀ x ∈ R

The minimum value of f is 0 and the point of minimum value of f is x = 0.

The graph of function f* *has no maximum value, and hence, no point of maximum value of f in R.

Let f be a function defined on an open interval D. Let f be continuous at a critical point, a, in D.

**Local minima**

If f '(x) changes its 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 the point of local minima.

The local minimum value is f(a).

**Local maxima**

If f '(x) changes its 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 the point of local maxima.

The local maximum value is f(a).

'0' is neither a point of local maxima nor a point of local minima.

'0' is the point of inflection.

**Point of inflection**

If f '(x) does not change its sign as x increases through critical point "a", then "a" is known as the point of inflection.

At point "a", f has neither a point of local maxima nor a point of local minima.

**Steps for finding local maxima and local minima:**

**Step1:** Find the derivative of f(x), i.e. f '(x).

**Step2:** Let a_{1},a_{2},a_{3},... ... ..., a_{n} be the roots of f '(x) = 0 . Here, a_{1},a_{2},a_{3},... ... ..., a_{n} are called the critical points or stationary points of x.

**Note:** The local maxima or local minima of f(x) can be attained at some or all these stationary points of x.

**Step 3:**Consider x = a_{1}. If f '(x) > 0 at the left neighbourhood values of a_{1} and if f '(x) < 0 at the right neighbourhood values of a_{1} , then x = a_{1} is the point of local maxima.

Consider x = a_{1}. If f '(x) < 0 at the left neighbourhood values of a_{1} and if f '(x) > 0 at the right neighbourhood values of a_{1}, then x = a_{1} is the point of local minima.

**Note:** If f '(x) does not change its sign as x increases through a_{1 }, then x = a_{1} is neither a point of local maxima nor a point of local minima.