# IB Mathematics HL SL-Maxima and Minima

In my previous post, we discussed how to find the equation of tangents and normal to a curve. There are a few more Applications of Derivatives in IB Mathematics HL SL, ‘Maxima and Minima’ is one of them.

**Maxima and Minima:-**

1. A function f(x) is said to have a maximum at x = a if f(a) is greater than every other value assumed by f(x) in the immediate neighbourhood of x = a. Symbolically

gives maxima for a sufficiently small positive h.

Similarly, a function f(x) is said to have a minimum value at x = b if f(b) is least than every other value assumed by f(x) in the immediate neighbourhood at x = b. Symbolically

If x = b gives minima for a sufficiently small positive h.

**►**The maximum & minimum values of a function are also known as local/relative maxima or local/relative minima as these are the greatest & least values of the function relative to some neighbourhood of the point in question.

**►** The term ‘extremum’ or (extremal) or** ‘turning value’** is used both for maximum or a minimum value.

**►** A maximum (minimum) value of a function may not be the greatest (least) value in a finite interval.

**►**A function can have several maximum & minimum values & a minimum value may even be greater than a maximum value.

**►**Maximum & minimum values of a continuous function occur alternately & between two consecutive maximum values, there is a minimum value & vice versa.

**2. A Necessary Condition For Maximum & Minimum:-** If f(x) is a maximum or minimum at x = c & if f'(c) exists then f'(c) = 0.

**►**The set of values of x for which f'(x) = 0 are often called as stationary points or critical points. The rate of change of function is zero at a stationary point. In IB Mathematics HL SL questions are asked on these points

**►**In case f'(c) does not exist f(c) may be a maximum or a minimum & in this case left hand and right-hand derivatives are of opposite signs.

**►** The greatest **(global maxima)** and the least** (global minima)** values of a function f in an interval [a, b] are f(a) or f(b) or are given by the values of x for which f'(x) = 0.

**►** Critical points are those where f'(x)= 0 if it exists, or it fails to exist either by virtue of a vertical tangent or by virtue of a geometrical sharp corner but not because of discontinuity of function.

**3. Sufficient Condition For Extreme Values:-**

x=c is a point of local maxima where f'(c)=0

similarly, x=c is a point of local maxima where f'(c)=0. Here ‘h’ is

sufficiently small positive quantity.

**► **If f'(x) does not change sign i.e. has the same sign in a certain complete neighbourhood of c, then f(x) is either strictly increasing or decreasing throughout this neighbourhood implying that f(c) is not an extreme value of the given function.

**4.Use Of Second Order Derivative In Ascertaining The Maxima Or Minima:-**

(a) f(c) is a minimum value of the function f, if f'(c) = 0 & f”(c) > 0.

(b) f(c) is a maximum value of the function f, if f'(c) = 0 & f”(c) < 0.

**►**If f'(c) = 0 then the test fails. Revert back to the first order derivative check for ascertaining the maxima or minima.

**►** If y = f make the function (x) is a quantity to be maximum or minimum, find those values of x for which f'(x)=0

**►**Test each value of x for which f'(x) = 0 to determine whether it provides a maximum or minimum or neither. The usual tests are :

(a) If is positive when y is minimum. If is negative when y is

maximum. If when the test fails.

(b) If is positive for then a maximum occurs at

(c)If is zero for then a maximum occurs at

(d)If is negative for then a maximum occurs at

(e) But if dy/dx changes sign from negative to zero to positive as x advances through xo there is a minimum. If dy/dx does not change sign, neither a maximum nor a minimum. Such points are called **Points of Inflection**.

Point of inflexion is a point where the shape of f(x) changes from concave to convex or convex to concave. This concept is usually asked in IB Mathematics HL SL

**►**If the derivative fails to exist at some point, examine this point as possible maximum or minimum.

**►**If the function y = f (x) is defined for only a limited range of values then we should examine x = a & x=b for possible extreme values.

**►**If the derivative fails to exist at some point, we should examine this point as possible maximum or minimum.

**Important Notes -:**

Given a fixed point

A(x_{1}, y_{1}) and a moving point P(x, f (x)) on the curve y = f(x). Then AP will be maximum or minimum if it is normal to the curve at P

**►**If the sum of two positive numbers x and y is constant than their product is maximum if they are equal, i.e. x + y = c , x > 0 , y > 0 , then xy = [(x + y)^{2} – (x – y)^{2}]

**►**If the product of two positive numbers is constant then their sum is least if they are equal.i.e. (x+y)^{2}=(x-y)^{2}+ 4xy

**6. Useful Formulae Of Mensuration To Remember**

a. The volume of a cuboid = lbh

b. Surface area of a cuboid = 2 (lb + bh + hl)

c. The volume of a prism = area of the base x height.

d. The lateral surface of a prism = perimeter of the base x height.

e. The total surface of a prism = lateral surface + 2 area of the base (Note that lateral surfaces of a prism are all rectangles) f.

f. The volume of a pyramid = area of the base x height

g. Volume of a cone =

h. The curved surface of a cylinder =

i. Total surface of a cylinder =

j. Volume of a sphere =

k. Surface area of a sphere =

l. Area of a circular sector =

### Click here to download questions on “Maxima and Minima”

### Worksheets on Maxima & Minima (22-09-15).pdf

Here are the links of some other posts on Application of derivatives

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