Need help with tricky question about functions

Question

I'm stuck on the following question, (particularly part b), can someone help me please? Have an exam tomorrow so help ASAP would be appreciated!

For the function f(x)=mx³ + x² + nx + p, where m, n, p are elements of {R}\0:

a) Find the x-coordinates of the turning points.

b) Express n in terms of m if f(x) has an inverse function.

Answer 1

Turning points on a graph are the points of concavity! Not the slope! Concavity is the 2nd derivative of the function d''f/dx which reads 2nd derivative of the function with respect to x.

So all you have to do is find the derivative then find the derivative of the derivative which is the 2nd derivative of the given function. Think of m, n, and p as just real numbers you can even paste in regular real numbers to make more sense of the problem!

1st derivative is : 3(m)(x^2) + 2(x) +0= 3(m)(x^2)+2(x)

2nd derivative is : 6(m)(x) +2

So it looks like there is only one point of concavity which means only 1 turning point!

set 2nd derivative = 0 to find turning point= point of concavity

6(m)(x) +2= 0 solve for x

x = -(1/3(m)) is the x coordinate of the turning point of this function

If the function has an inverse means that there exists a f(f(x)) that is in respect to the given f(x) function

To find this solve for x in the given function!

call the initial function f(x) and set it equal to y in the given equation

and solve for x so you get a f(y) which is a function with respect to y