https://www.quora.com/What-is-the-difference-between-local-minima-maxima-and-absolute-minima-maxima
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Engineering Mathematics-I Maxima and Minima Prepared by : Prof. Rupali Yeole Hope Foundation’s International Institute of Information Technology, I²IT, P-14 Rajiv Gandhi Infotech Park, Hinjawadi, Pune - 411 057 Tel - +91 20 22933441 / 2 / 3 | Website - www.isquareit.edu.in ; Email - info@isquareit.edu.in
Maxima and Minima of Functions of Two Independent Variables • Let f (x, y) be a function of two independent variables x and y , which is continuous for all values of x and y in the neighborhood of a, b i.e. a h, b k be a point in its neighborhood which lies inside the region R .
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• The point a, b is called a point of relative minimum, if f (a, b) f (a h, b k) for all h, k Then f (a, b) is called the relative minimum value. • The point a, b is called a point of relative maximum, if f (a, b) f (a h, b k) for all h, k Then f (a, b) is called the relative minimum value.
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• Stationary point: The point at which function is either maximum or minimum is known as stationary point. • Extreme Value: The value of the function at stationary point is known as extreme value of the function f (x, y) .
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Working Rule To determine the maxima and minima (extreme values) of a function f (x, y) • Step I : Solve f f 0& 0 simultaneously for x & y x
y
• Step II: Obtain the values of 2 f 2 f 2 f r 2 , s , t 2 x xy y Hope Foundation’s International Institute of Information Technology, I²IT, P-14 Rajiv Gandhi Infotech Park, Hinjawadi, Pune - 411 057 Tel - +91 20 22933441 / 2 / 3 | Website - www.isquareit.edu.in ; Email - info@isquareit.edu.in
• Step III: 1) If rt s 2 0 & r 0 at a, b , then f (x, y) is maximum at a, b & maximum value of the function is f (a, b). 2) If rt s 2 0 & r 0 at a, b , then f (x, y) is maximum at a, b & maximum value of the function is f (a, b) . 3) If rt s 2 0 at a, b , then f (x, y) is neither maximum nor minimum at a, b .Such point is called Saddle Point. 4) If rt s 2 0 at a, b , then no conclusion can be made about the extreme values of f (x, y) & further investigation is required. Hope Foundation’s International Institute of Information Technology, I²IT, P-14 Rajiv Gandhi Infotech Park, Hinjawadi, Pune - 411 057 Tel - +91 20 22933441 / 2 / 3 | Website - www.isquareit.edu.in ; Email - info@isquareit.edu.in
Example 1
Q.1 Discuss the maxima and Minima of the function x y 6 x 12 Answer: Let f x, y x 2 y 2 6 x 12 Step I: For extreme values 2
2
f 0 x 2 x 6 0, 2( x 3) 0 x 3 f & 0 y 2 y 0, y 0 Hope Foundation’s International Institute of Information Technology, I²IT, P-14 Rajiv Gandhi Infotech Park, Hinjawadi, Pune - 411 057 Tel - +91 20 22933441 / 2 / 3 | Website - www.isquareit.edu.in ; Email - info@isquareit.edu.in
Therefore stationary point is 3,0 . • Step II: 2 f r 2 2, x 2 f s 0, xy 2 f t 2 2. y Hope Foundation’s International Institute of Information Technology, I²IT, P-14 Rajiv Gandhi Infotech Park, Hinjawadi, Pune - 411 057 Tel - +91 20 22933441 / 2 / 3 | Website - www.isquareit.edu.in ; Email - info@isquareit.edu.in
• Step III: At 3,0 rt s 2 2 2 0 4 0 & r20
Hence f (x, y) is maximum at 3,0 f min 3 0 6 3 12 2
2
3
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THANK YOU For further details please contact RUPALI YEOLE Department of Applied Sciences & Engineering Hope Foundation’s International Institute of Information Technology, I²IT P-14, Rajiv Gandhi Infotech Park, MIDC Phase 1, Hinjawadi, Pune – 411 057 www.isquareit.edu.in Phone : +91 20 22933441 / 2 / 3 rupaliy@isquareit.edu.in | info@isquareit.edu.in