CAT Quantitative Aptitude Questions | CAT Algebra - Inequalities questions

CAT Questions | Algebra | Inequalities - Minimum Value

The question is about minimizing a function. Our task is to find out the minimum value of a quadratic function. Do we need calculus for that? Inequalities are crucial to understand many topics that are tested in the CAT exam. Having a good foundation in this subject can help us tackle questions in Coordinate Geometry, Functions, and most importantly in Algebra. A range of CAT questions can be asked based on this simple concept.

Question 21: What is the minimum value of f(x) = x2 โ€“ 5x + 41?

  1. \\frac{139}{4}\\)
  2. \\frac{149}{4}\\)
  3. \\frac{129}{4}\\)
  4. \\frac{119}{4}\\)

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Explanatory Answer

Method of solving this CAT Question from Algebra - Inequalities : Show this question to anyone who says you need differentiation to crack CAT exam.

This question is present for one reason and one reason only. To talk about the idea of โ€œcompletion of squaresโ€?. There are two ways of solving this question
The ugly differentiation based method and the beautiful completion of squares method.
Always pick the elegant method. You might not prefer VVS Laxman over Gary Kirsten, or Federer over Nadal. But these are matters of sport. When it comes to math solutions โ€“ elegant solutions kick ass every time.

What is this famous completion of squares method ?
Any quadratic expression of the form x2 + px + q can be written in the form (x +a)2 + b.
Write in that form, enjoy the equation and have some fun.
x2 โ€“ 5x + 41 = (x + a)2 + b. what value should โ€˜aโ€™ take? Forget about b for the time being.

(x + a)2 = x2 + 2ax + a2. The 2ax term should correspond to -5x. Done and dusted.
a =\\frac{-5}{2}\\)a2 = \\frac{25}{4}\\)
x2 โ€“ 5x + 41 can be written as x2 โ€“ 5x + \\frac{25}{4}\\) โ€“ \\frac{25}{4}\\) + 41 = x โ€“ \\frac{5}{2}\\)2 + 41 โ€“ \\frac{25}{4}\\)
= x โ€“ \\frac{5}{2}\\)2 + \\frac{139}{4}\\)
The minimum value this expression can take is \\frac{139}{4}\\)

The question is "What is the minimum value of f(x) = x2 โ€“ 5x + 41?"

Hence the answer is "\\frac{139}{4}\\)"

Choice A is the correct answer.

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