The sum of the maximum and minimum values of the function f(x)=|5 x-7|+[x^2+2 x] in the interval [(5 / 4), 2], where [t] is the greatest integer <= t, is ______________.

f(x) = |5x - 7| + [x^2 + 2x] = |5x - 7| + [(x + 1)^2] - 1 Critical points of f(x) = 7 / 5,sqrt 5 - 1, sqrt 6 - 1, sqrt 7 - 1, sqrt 8 - 1, 2 Therefore Maximum or minimum value of f(x) \ occur at critical points or boundary points Therefore f( 5 / 4 ) = 3 / 4 + 4 = 19 / 4 f( 7 / 5 ) = 0 + 4 = 4 as both |5x - 7| and x^2 + 2x \ are increasing in nature after x = 7 / 5 Therefore f(2) = 3 + 8 = 11 Therefore f( 7 / 5 )_min = 4 \ and f(2)_max = 11 Sum is 4 + 11 = 15

The sum of the maximum and minimum values of the functi... | null (Mathematics)

f(x) = |5x - 7| + [{x^2} + 2x]

= |5x - 7| + [{(x + 1)^2}] - 1

\text{Critical points of}

f(x) = {7 \over 5},\sqrt 5 - 1,\,\sqrt 6 - 1,\,\sqrt 7 - 1,\,\sqrt 8 - 1,\,2

\therefore \text{Maximum or minimum value of} \ f(x) \\ \text{occur at critical points or boundary points}

\therefore f\left( {{5 \over 4}} \right) = {3 \over 4} + 4 = {{19} \over 4}

f\left( {{7 \over 5}} \right) = 0 + 4 = 4

\text{ as both } |5x - 7| \text{ and } {x^2} + 2x \\ \text{ are increasing in nature after } x = {7 \over 5}

\therefore f(2) = 3 + 8 = 11

\therefore f{\left( {{7 \over 5}} \right)_{\min }} = 4 \\\\ \text{and} \ f{(2)_{\max }} = 11

\text{Sum is} \ \ 4 + 11 = 15

Explanation

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