Find the stable points of the following function, determine wether or not they are maxima or minima. y= 5x^3 +9x^2 +3x +2

Start by differentiating the function to find points where the gradient is 0. so dy/dx = 15x^2 + 18x + 3 We can use the equation for finding the roots of a quadratic here, set a=15, b=18 and c=3 and proceed. You will get two roots, one at x= -1 and one at x=-(1/5).

So we have two stable points, to find their corresponding values of y, simply plug these values back into the original equation (y=5x^3 +9x^2 +3x + 2). When x=-(1/5) y=1.72 and when x=-1 y=3. Thus we have found the two stable points.

Finally, to find whether or not these points are maxima or minima we differentiate the function again, this is called the double differential and will tell us if the value for the gradient is increasing or decreasing.

so (d^2y/dx^2) = 30x + 18 when x= -1/5 (d^2y/dx^2)= 12. This means that the value of the gradient is increasing at this point, hence this is a minima.

When x=-1 (d^2y/dx^2)= -12 This means that the value for the gradient is decreasing at this point, hence this is a maxima.