IB Mathematics HL SL-Maxima and Minima

In the previous post, IB Maths Tutors 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

\left. \begin{array}{l} f(a) > f(a + h)\\ f(a) > f(a - h) \end{array} \right] \Rightarrow x = a   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

\left. \begin{array}{l} f(b) > f(b + h)\\ f(b) > f(b - h) \end{array} \right]  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 neighborhood 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 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

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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:-

\left. \begin{array}{l} f'(c - h) > 0\\ f'(c + h) < 0 \end{array} \right]  \Rightarrow x=c is a point of local maxima where f'(c)=0

similarly,  \left. \begin{array}{l} f'(c - h) < 0\\ f'(c + h) > 0 \end{array} \right] \Rightarrow  x=c is a point of local maxima where f'(c)=0. Here ‘h’ is

a 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 neighborhood 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.

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 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     \frac{{{d^2}y}}{{d{x^2}}} is positive when \frac{{dy}}{{dx}} = 0  y is minimum. If \frac{{{d^2}y}}{{d{x^2}}}  is negative when  \frac{{dy}}{{dx}} = 0  y is

maximum. If  \frac{{{d^2}y}}{{d{x^2}}} when  \frac{{dy}}{{dx}} = 0 the test fails.

(b) If   \frac{{dy}}{{dx}} is positive for  x > {x_0} then a maximum occurs at  x = {x_0}

(c)If   \frac{{dy}}{{dx}} is zero for   x = {x_0} then a maximum occurs at  x = {x_0}

(d)If   \frac{{dy}}{{dx}} is negative  for    x > {x_0}  then a maximum occurs at  x = {x_0}

(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  a \le x \le b  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.

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Important Notes -:
Given a fixed point

A(x1, y1) 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 = \frac{1}{3}\pi {r^2}h

h. The curved surface of a cylinder =  2\pi rh

i. Total surface of a cylinder =  2\pi r(h + r)

j. Volume of a sphere = \frac{4}{3}\pi {r^3}

k. Surface area of a sphere =  4\pi {r^2}

l. Area of a circular sector =   \frac{{\pi {r^2}\theta }}{{{{360}^0}}}

Click here to download questions on “Maxima and Minima”

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

Tangents And Normals

Increasing and Decreasing


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