Thread: Differential Calculus Problem on a family of cubic functions

Hi,

I got this problem on a math test that I (unfortunately) failed.
A family of cubic functions is defined as f(x)= (k^2)*(x^3) - k(x^2)+x where k is an element of the set of positive integers.

a) Express in terms of k
i) the first and second derivative of f(x)
ii) the coordinates of the points of inflexion P on the graphs of f

b) show that all P lie on a straight line and state its equation
c) Show that for all values of k the tangents to the graph of f at P are parallel and find the equation of the tangent lines.

I have no idea what to do for b. This is what I need help with. I need to relearn this before the retake, so answers are of course appreciated.

I solved a and c, and they were correct.

a) first derivative: 3(k^2)*(x^2) - 2kx +1
second derivative: 6(k^2)x - 2k
ii) when Second derivative = 0 x is equal to (1/3k). When x = (1/3k) y = (7/27k)

c) The slope of the tangent at P is the value of the first derivative at P. The slope is 2/3 which is a constant that is the same for all values of k. The equation can be found by substituting into y=mx+c the value for m and y and x that we know and solving for c. (7/27k)= (2/3)*(1/3k) + c therefore c is 1/27k and the equation is y=(2/3)x+ (1/27k)

The only part I really don't understand is B, but I included my work in case it would be helpful. I've tried to be as clear as possible, if anyone has ideas on how to make my math clearer, help is always appreciated. Hopefully somebody can explain this to me. The only thing I would ask is that the explanation is in terms suitable for the skills I've learned so far, as I am in high school. (IB HL if anyone knows what that is.)I assume it's a relatively simple thing since part b is only worth 2/13 points in the entire exercise. Thank you very much in advance for the help!

Originally Posted by globalfaerie

Hi,

I got this problem on a math test that I (unfortunately) failed.
A family of cubic functions is defined as f(x)= (k^2)*(x^3) - k(x^2)+x where k is an element of the set of positive integers.

a) Express in terms of k
i) the first and second derivative of f(x)
ii) the coordinates of the points of inflexion P on the graphs of f

b) show that all P lie on a straight line and state its equation
c) Show that for all values of k the tangents to the graph of f at P are parallel and find the equation of the tangent lines.

I have no idea what to do for b. This is what I need help with. I need to relearn this before the retake, so answers are of course appreciated.

I solved a and c, and they were correct.

a) first derivative: 3(k^2)*(x^2) - 2kx +1
second derivative: 6(k^2)x - 2k
ii) when Second derivative = 0 x is equal to (1/3k). When x = (1/3k) y = (7/27k)

Be careful how you write those. I think you mean x= 1/(3k) and y= 7/(27k). So for B you just need to show that these points all lie along some straight line. One good way to do that is to show that the slopes are all the same. Actually, here, you can say more- they all lie along a straight line through the origin. Show that the slopes of the lines through (0, 0) and (1/(3k), 7/(27k)) are all the same.

c) The slope of the tangent at P is the value of the first derivative at P. The slope is 2/3 which is a constant that is the same for all values of k. The equation can be found by substituting into y=mx+c the value for m and y and x that we know and solving for c. (7/27k)= (2/3)*(1/3k) + c therefore c is 1/27k and the equation is y=(2/3)x+ (1/27k)

The only part I really don't understand is B, but I included my work in case it would be helpful. I've tried to be as clear as possible, if anyone has ideas on how to make my math clearer, help is always appreciated. Hopefully somebody can explain this to me. The only thing I would ask is that the explanation is in terms suitable for the skills I've learned so far, as I am in high school. (IB HL if anyone knows what that is.)I assume it's a relatively simple thing since part b is only worth 2/13 points in the entire exercise. Thank you very much in advance for the help!