Let the range of the function f(x) = c x - 3 x + d , where c and d are positive real constants, be [1, 3] . The value of c + d is equal to

Step-by-step solution

The given function is f(x) = c x - 3 x + d . The expression x - 3 x is of the form A x + B x , whose range is [- A^2+B^2 , A^2+B^2 ] . Here, the range of x - 3 x is [- 1^2+(- 3 )^2 , 1^2+(- 3 )^2 ] = [-2, 2] . Therefore, the range of the denominator x - 3 x + d is [d-2, d+2] . Since c > 0 and the range of f(x) is strictly positive ( [1, 3] ), the denominator must be strictly positive, meaning d - 2 > 0 . The maximum value of f(x) occurs when the denominator is minimum, and the minimum value of f(x) occurs when the