For the quartic function f(x)=x^4-4x^3+8x+5, determine algebraically the:

a) equation of line PS
b) equation of the tangent to the function at Q
c) coordinates of point M
e) areas of sections marked A, B, C and D
f) write the ratio of areas A:B:C:D in the form 1:b:c:d

Question

For the quartic function f(x)=x^4-4x^3+8x+5, determine algebraically the:

a) equation of line PS 
b) equation of the tangent to the function at Q
c) coordinates of point M
e) areas of sections marked A, B, C and D
f) write the ratio of areas A:B:C:D in the form 1:b:c:d

anonymous · Answer

attachment

mathmale · Answer

Interesting problem!  What have you thought of doing first?  Note that there's no area D in your illustration.  Perhaps I need new glasses?

mathstudent55 · Answer

Area D = Area B + Area B2
It states so in the figure.

mathstudent55 · Answer

You can find inflection points by finding the second derivative of the function and setting it equal to zero. Then solve for x. The roots of this equation are the x-coordinates of the inflection points. Plug in the x-coordinates in to the original function to find the corresponding y-coordinates. This will give you the coordinates of the inflection points, Q and R.

1. The equation of line PS is the equation of line QR whose points you now have.
2. The slope of the tangent is found by taking the first derivative of the function (which you already have, since you needed to take the first derivative to get to the second derivative). Then plug in the x-coordinate of Q. Now that you have the slope, use the slope and the coordinates of point Q to find the equation of the tangent.