Find the stationary point on the line of y = 6x - x^2 and state whether this point is a maximum or a minimum

This question requires differentiating the equation to find where its gradient is equal to zero. Differentation is done via a simple equation -->  if y = xⁿ then dy/dx = nx^n-1

Therefore if y = 6x - x²  , then dy/dx = 6(x⁰) - 2x¹   , so   dy/dx = 6 - 2x

The gradient is 0 at the stationary point, so 6 - 2x = 0           2x = 6        so x = 3

To find y, substitute (x=3) into the original formula to find y.     y = 6(3) - 3²   = 18 - 9    = 9

The stationary point is (3,9), and to find out whether this is a maximum or minimum, x=4 can be subbed in to the formula to find the next point on the line. y = 6(4) - 4²   =  24 - 16     =    8     so the next point is (4,8)

This is below the stationary point, so we can see that (3,9) is a maximum.