Question

The numerator of a fraction is multiple of two numbers. One of the numbers is greater than the other by 2. The greater number is smaller than the denominator by 4. If the denominator 7+C (C > 7) is a constant, then the minimum value of the fraction is

Answer 1

denominator  =  7+C     (given  C > - 7 and C is a constant)
numerator =  (C+3) (C+1)

Fraction = F = (C+1)(C+3) / (C+7) = C - 3 + 24/(C+7)

Derivative dF/dC = 1 - 24/(C+7)² = 0    =>  C+7 = 2√6
2nd derivative d²F/dC² =  48/(C+7)³  > 0   as   C > -7

So F is minimum at  C+7 = 2√6,
      F minimum = (C - 3) + 24/(C+7) = 2√6 - 7 - 3  +  24/(2√6)
                          = 4 √6 - 10 = - 0.2020