How do I find the coordinates of maximum and minimum turning points of a cubic equation?

Finding a maximum and minimum involves differentiating the function in order to determine when the gradient is at zero as this will be when the function has its maximum and minimum turning points.

take the function: f(x)=x3+6x2+9x+2

we must first obtain the gradient function through differentiation

d/dx = 3x2+12x+9 using standard differentiation technique (multiplying the power of the unknown by its coefficient to generate its new coefficient and then reduce its power by one to find the new term)

now that we have the gradient function it must be set to zero, since the gradient of the function is zero at a turning point


this is a quadratic and we must solve for x to find the x values at which the gradient is zero

by inspection:

(3x+3)(x+3)=0 and therefore when the gradient of the function is zero:

x=-1 and x=-3

to determine which of these is maximum and is which is minimum we can differentiate the gradient function again. This generates a function that tells us the rate of change of the gradient at any given value of x


d2/dx2=6x+12 (using the same method as before)

if we substitue in our two x values:

when x=-1 d2/dx2= -6+12 = 6

since this is positive, the gradient is changing in a positive direction and so when x= -1 it is a minimum.

when x=-3 d2/dx2= -18+12 = 6

since this is negative the gradient is changing in a negative direction and so when x= -3 its a maximum.

to find the y values of these x values we simply subsitute them into the original function:




so our maximum is (-3,2)

and our minimum is (-1,-2)

Matthew K. GCSE Maths tutor, GCSE Physics tutor

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