{"id":465,"date":"2017-09-28T23:45:51","date_gmt":"2017-09-28T18:15:51","guid":{"rendered":"http:\/\/ibelitetutor.com\/blog\/?p=465"},"modified":"2023-08-14T12:53:26","modified_gmt":"2023-08-14T07:23:26","slug":"maxima-and-minima-problems","status":"publish","type":"post","link":"http:\/\/ibelitetutor.com\/blog\/maxima-and-minima-problems\/","title":{"rendered":"IB Mathematics HL SL-Maxima and Minima"},"content":{"rendered":"

In the previous post, IB Maths Tutors<\/strong> discussed how to find the equation of tangents and normal to a curve. There are a few more \u00a0Applications of Derivatives in IB Mathematics HL SL, ‘Maxima and Minima’\u00a0is one of them.<\/p>\n

Maxima and Minima:-<\/strong><\/p>\n

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<\/p>\n

\"\\left.\u00a0\u00a0\u00a0gives maxima for a sufficiently small positive h.<\/p>\n

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<\/p>\n

\"\\left.\u00a0\u00a0If x = b gives minima for a sufficiently small positive h.<\/p>\n

<\/p>\n

\u25ba<\/strong>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.<\/p>\n

\u25ba<\/strong> The term ‘extremum’ or (extremal) or ‘turning value’<\/strong> is used both for maximum or a minimum value.<\/p>\n

\u25ba<\/strong> A maximum (minimum) value of a function may not be the greatest (least) value in a finite interval.<\/p>\n

\u25ba<\/strong>A\u00a0function can have several maximum & minimum values & a minimum value may even be greater than a maximum value.<\/p>\n

\u25ba<\/strong>Maximum & minimum values of a continuous function occur alternately & between two consecutive maximum values, there is a minimum value & vice versa.<\/p>\n

2. A Necessary Condition For Maximum & Minimum:-<\/strong> If f(x) is a maximum or minimum at x = c & if f'(c) exists then f'(c) = 0.<\/p>\n

\u25ba<\/strong>The set of values of x for which f'(x) = 0 are often called stationary points<\/span> or critical points<\/span>. The rate of change of function is zero at a stationary point. In IB Mathematics HL SL questions are asked on these points<\/p>\n

\u25ba<\/strong>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.<\/p>\n

\u25ba<\/strong> The greatest (global maxima)<\/strong> and the least (global minima)<\/strong> 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.<\/p>\n

\u25ba<\/strong> 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.<\/p>\n

3. Sufficient Condition For Extreme Values:-<\/strong><\/p>\n

\"\\left.\u00a0\"x=c is a point of local maxima where f'(c)=0<\/p>\n

similarly, \u00a0\"\\left.\u00a0x=c is a point of local maxima where f'(c)=0. Here ‘h’ is<\/p>\n

a sufficiently small positive quantity.<\/p>\n

\u25ba\u00a0<\/strong>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.<\/p>\n

4.Use Of Second Order Derivative In Ascertaining The Maxima Or Minima:-<\/strong><\/p>\n