Minimum problem 2

Algebra Level pending

If $x^2+4+3\cos (ax+b) = 2x$ , find the minimum value of $\lfloor | a+b | \rfloor$ .

1 solution

Prajwal Krishna
Nov 22, 2016

${ x }^{ 2 }-2x+1+3\quad =\quad -3\cos { (ax+b) }$

$({ x }-1)^{ 2 }+3\quad =\quad -3\cos { (ax+b) }$

LHS minimum value = 3 when x=1

RHS maximum value is 3 when cos (ax+b) = -1

Therefore both these has to satisfy simultaneously

Cos (a+b) = -1

Therefore minimum value of a+b is $\Pi$

$\left\lfloor \pi \right\rfloor$ = 3