Maxima and minima problems are frequently encountered in competitive exams, particularly in algebra and geometry. These types of problems require finding the maximum or minimum value of a given function or expression subject to certain constraints. In algebra, the constraints might involve equations or inequalities, while in geometry, they might involve geometric shapes such as circles or triangles. Solving maxima and minima problems requires a solid understanding of calculus and optimization techniques.
Multiple Choice Questions
Que 1: Find the maximum value of 3x+4y if x²+y²=9
(a) 15
(b) 12
(c) 7
(d) None
Solution: Here we use Cauchy-Schwarz Inequality
(ab+cd)^2\leq (a^2+c^2)(b^2+d^2).
(3b+4d)^2\leq (3^2+4^2)(b^2+d^2).
(3b+4d)^2\leq 25\times 9.
(3b+4d)\leq 15.
Option (a) is correct.
Que 2: Find the minimum value of \sqrt{x^2-4x+13} +\sqrt{x^2-14x+130}.
(a) 7
(b) 9√2
(c) 13
(d) 6√3
Solution: \sqrt{(x^2-4x+4)+9}+\sqrt{(x^2-14x+49)+81}.
⇒\sqrt{(x-2)^2+3^2}+\sqrt{(7-x)^2+9^2}.
Here we use Vector Triangle Inequality: (Maxima and minima problems)
\sqrt{a^2+b^2}+\sqrt{c^2+d^2}\geq\sqrt{(a+c)^2+(b+d)^2}.
\sqrt{(x-2)^2+3^2}+\sqrt{(7-x)^2+9^2}\geq\sqrt{(x-2+7-x)^2+(3+9)^2}.
\geq\sqrt{5^2+12^2}.
\geq 13.
Option (c) is correct.
Que 3: Find the minimum value of a²+b²+c² if a, b, and c are the sides of a triangle of a given area of 7 cm².
(a) 14
(b) 21√2
(c) 28√3
(d) None
Solution: Here we use Weitzenbock’s Inequality
If a, b, and c are the lengths of the sides of a triangle and △ represents the area of the triangle.
Then, a2+b2+c2≥4√3.△
a2+b2+c2≥4√3×7
a2+b2+c2≥28√3
Option (c) is correct.
