y = x³ + ax² + bx + 6

dy/dx = 3x² + 2ax + b

For y to be a decreasing function of x, it means that as x increases, y decreases.

So the curve's gradient is always negative. i.e dy/dx < 0

So,

3x² + 2ax + b < 0

Since the question says the only values of x are 2 < x < 9 , and we have a positive coefficient of x² above,

(meaning that the quadratic curve is U-shaped , upward sloping)


We can say that (x - 2)(x - 9) < 0


Expanding this,


x² - 9x - 2x + 18 < 0

x² - 11x + 18 < 0

Multiply throughout by 3,

3x² - 33x + 54 < 0


Comparing coefficients with 3x² + 2ax + b < 0

2a = -33, b = 54
a = -33/2 (or -16½)


There is an error in the question, a cannot be an integer