# CAT Practice : Inequalities

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## Inequalities - Minimum Value

Q.21:What is the minimum value of f(x) = x2 – 5x + 41?
1. ${\frac{139}{4}}$
2. ${\frac{149}{4}}$
3. ${\frac{129}{4}}$
4. ${\frac{119}{4}}$

Choice A. ${\frac{139}{4}}$

## Detailed Solution

This question is present for one reason and one reason only. To talk about the idea of “completion of squares”. 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 – 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 – 5x + 41 = (x + a)2 + b. what value should ‘a’ 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 – 5x + 41 can be written as x2 – 5x + $( \frac{25}{4} )$$( \frac{25}{4} )$ + 41 = (x – $( \frac{5}{2} )$)2 + 41 – $( \frac{25}{4} )$

= (x – $( \frac{5}{2} )$)2 + ${\frac{139}{4}}$

The minimum value this expression can take is ${\frac{139}{4}}$.

Correct Answer: B. ${\frac{139}{4}}$

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## More questions from Inequalities

Inequalities are crucial to understand many topics that are tested in the CAT. Having a good foundation in this subject will make us tackling questions in Coordinate Geometry, Functions, and most importantly in Algebra much more comfortable.