For the function f(x) = 4x^3 -3x^2 - 6x, find a) All points where df/dx = 0 and b) State if these points are maximum or minimum points.
Part a) requires you to find df/dx for the given function. To do this, we differentiate the function once, which is done by multiplying the power of each 'part' with the part itself and subtracting 1 from the power. So, 4x3 ---> 12x2 , -3x2 ---> -6x and -6x ---> -6, giving df/dx = 12x2 -6x -6.Then, we simply solve this to equal zero, which we can do through simplifying, then factorising: 12x2 -6x-6 = 0 ---> 2x2 -x -1 = 0 (2x +1)(x -1) = 0, therefore x = 1, x = -1/2 are both solutions to this. We then substitute these values into f(x) to determine our full coordinate values, giving our answer to part a) as (1, -5) and (-1/2, 7/4) Part b) asks for maxima and minima, requires us to find d2f/dx2 which can be found by differentiating df/dx using the same method as before. This gives us d2f/dx2 = 24x - 6. Now, a maximum point of a graph is when d2f/dx2 < 0, and a minima occurs when d2f/dx2 > 0. As such, we simply substitue the values we found from part a) into our new equation to determine our new answers: at x = 1, d2f/dx2 = 24 - 6 = 18 > 0, therefore x = 1 is a minima; at x = -1/2, d2f/dx2 = -12 -6 = -18 < 0, therefore x = -1/2 is a maxima.
