A curve has equation -2x^3 - x^2 + 20x . The curve has a stationary point at the point M where x = −2. Find the x-coordinate of the other stationary point of the curve.

Lets think about what stationary points actually mean. A stationary point of a curve is where the gradient is equal to 0. Illustrate by plotting the graph of y = x^3.We know that by differentiating the equation of a curve we get the gradient. If we then set that differential equation to 0 we can get the co-ordinates of the stationary point. dy/dx = 0 at the stationary points. Emphasise the importance of writing this statement down. Now differentiate the equation to get -6x^2 - 2x + 20. Set equation = 0 to get -6x^2 - 2x + 20 = 0. We can immediately simplify by dividing across by 2 to get -3x^2 - x + 10 = 0Its much easier to break the equation down into smaller terms can then be set = 0, this is called factorising. We also already know that one of these terms can be solved to give x = -2, which, when re-arranged gives x + 2 = 0. We now know that x + 2 multiplied by a mystery term gives -3x^2 - x + 10. We can therefore do a division to get our mystery term. do the division: -3x^2 - x + 10 divided by x + 2 to yield -3x + 5 as our mystery term. We can set this mystery term = 0 and re-arrange to get 3x = 5. We want x on its own which is therefore x = 5/3 after we divide across by 3. 5/3 is our answer.