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A function f(x) is said to have a local maximum at x = a if the value of f(a) is greater than all the values of f(x) in a small neighbourhood of x = a. Mathematically, f (a) > f (a – h) and f (a) > f (a + h) where h > 0, then a is called the point of local maximum.

A function f(x) is said to have a local minimum at x = a, if the value of the function at x = a is less than the value of the function at the neighboring points of x = a. Mathematically, f (a) < f (a – h) and f (a) < f (a + h) where h > 0, then a is called the point of local minimum.

- A point of local maximum or a local minimum is also called a point of local extremumA point where the graph of function is continuous and has a tangent line and where the concavity changes is called point of inflexion.
- At the point of inflexion, either y” = 0 and changes sign or y” fails to exist.
- At the point of inflexion, the curve crosses its tangent at that point.
- A function cannot have point of inflexion and extrema at the same point.
- Working rules to find points of local maxima and local minima:

**1. First Derivative Test:** If f'(a) = 0 and f'(x) changes its sign while passing through the point x = a, then

- f(x) would have a local maximum at x = a if f'(a – 0) > 0 and f'(a + 0) < 0. It means that f'(x) should change its sign from positive to negative.
- f(x) would have local minimum at x = a if f'(a – 0) < 0 and f'(a + 0) > 0 . It means that f'(x) should change its sign from negative to positive.
- If f(x) doesn’t change its sign while passing through x = a, then f (x) would have neither a maximum nor minimum at x = a. e.g. f (x) = x
^{3}doesn’t have any local maxima or minima at x = 0.

**2. ****Second Derivative Test:**

- Let f(x) be a differentiable function on a given interval and let f'' be continuous at stationary point. Find f'(x) and solve the equation f'(x) = 0 given let x = a, b, … be solutions.
**There can be two cases:**

Case (i): If f''(a) <0 then f(a) is maximum.

Case (ii): If f ''(a) > 0 then f(a) is minimum.

In case, f''(a) = 0 the second derivatives test fails and then one has to go back and apply the first derivative test.

If f''(a) = 0 and a is neither a point of local maximum nor local minimum then a is a point of inflection.

**3. n ^{th} Derivative Test for Maxima and Minima: **Also termed as the generalization of the second derivative test, it states that if the n derivatives i.e. f '(a) = f''(a) = f'''(a) =………. = f

- In some questions involving determination of maxima and minima, it might become difficult to decide whether f(x) actually changes its sign while passing through x = a and here, n
^{th}derivative test can be applied. - Global Minima & Maxima of f(x) in [a, b] is the least or the greatest value of the function f(x) in interval [a, b].

1. The function f(x) has a global maximum at the point ‘a’ in the interval I if f (a) ≥ f(x), for all x ∈ I.

2. Function f(x) has a global minimum at the point ‘a’ if f (a) ≤ f (x), for all x ∈ I.

- Global Maxima Minima always occur either at the critical points of f(x) within [a, b] or at the end points of the interval.
**Computation of Global Maxima and minima in maxima minima problems:**

1. Compute the critical points of f(x) in (a, b). Let the various critical points be C_{1}, C_{2}, …. , C_{n}.

2. Next, compute the value of the function at these critical points along with the end points of the domain. Let us denote these values by f(C_{1}), f(C_{2})………..f(C_{n}).

3. Now, compute M* = max{f(a), f(C_{1}), f(C_{2})………..f(C_{n}), f(b)} and M** = min{f(a), f(C_{1}), f(C_{2})………..f(C_{n}), f(b)}.Now M* is the maximum value of f(x) in [a, b] and M** is the minimum value of f(x) in [a, b].- In order to find global maxima or minima in open interval (a, b) proceed as told above and after the first two steps, compute

M_{1}= max{f(C_{1}), f(C_{2})………..f(C_{n})} and

M_{2 }= min{f(C_{1}),f(C_{2})………..f(C_{n})}. - Now if x approaches a- or if x approaches b- , the limit of f(x) > M
_{1}or its limit f(x) < M_{1 }would not have global maximum (or global minimum) in (a, b) but if as x approaches a- and x approaches b- , lim f(x) < M_{1}and lim f(x) > M_{2}, then M_{1 }and M_{2}will respectively be the global maximum and global minimum of f(x) in (a,b). - If f(x) is a continuous function on a closed bounded interval [a,b], then f(x) will have a global maximum and a global minimum on [a,b]. On the other hand, if the interval is not bounded or closed, then there is no guarantee that a continuous function f(x) will have global extrema.
- If f(x) is differentiable on the interval I, then every global extremum is a local extremum or an end point extremum.

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