Verified answer

To find equation of curve

3x^2 - 9x = gradient (it has been differentiated)

Turn it back

y = x^3 - 4.5x^2 + c

y = - 18, x = - 2 , plug these into what you have so far

- 18 = (- 2)^3 - 4.5( - 2)^2 + c

- 18 = - 8 - (4.5 * 4) + c

- 18 = - 8 - 18 + c

- 18 +18 + 8 = c

....8 = c

Equation of curve is

y = x^3 - (9/2x)^2 + 8

Turning points

Set Derivative to = 0

3x^2 - 9x = 0 (Factorise)

3x( x - 3) = 0

x = 3 and x = 0

So graph has 2 stationary points at x = 3 and x = 0

At x = 3 y = x (3)^3 - 9/2(3)^2 + 8

At x = 3, y= 27 - 40.5 + 8 = - 5.5

(3, - 5.5)

At x = 0, y = 0 + 0 + 8

(0, 8)

To find maximum and minimum differentiate again

dy/dx = 3x^2 - 9x

d2y / dx2 = 6x - 9 (stick in the coordinates of the stationary points)

At x = 3... d2y/dx2 ..18 - 9 = 9 which is positive so is a maximum

At x = 0 ...d2y/dx2...0 - 9 = - 9 which is negative so is a maximum

dy/dx = 3x^2-8x+5 a million. combine: the anti-spinoff/y = x^3 - 4x^2 + 5x + c 2. discover c Sub in (0,3): 3 = 0 - 0 + 0 + c c = 3 3. sub c into equation y = x^3 - 4x^2 + 5x + 3

f ' = 3x² - 9x

f = x^3 - (9/2) x^2 + c

-18 = -8 - (9/2)*4 + c

c = 8

f(x) = x^3 - (9/2) x^2 + 8

Regards - Ian