Expert: Bobby Soltani - 1/1/2005

Question
Let f be the function given by f(x)=3x^4+x^3-21x^2.

a.) write an equation of the line tangent to the graph of f at the point (2,-28).

b.) find the absolute minimum value of f.

c.) find the x-coordinate of each point of inflection on the graph of f.

Answer
Hi Stefanny,

There are a lot of steps to this problem so I will give you some tips on solving each part:

1.  to get the equation of a line, you need a point (which we have) and a slope.  To find the slope of the function at that point, take the derivative and find the value at f'(2).  Then use the slope and the point to find the equation of the line.

2.  Set the derivative of f equal to zero.  These are the x values of the minumums.  Plug these x values into f to find the corresponding y values.  Which ever value is the smallest, that is the minimum.

3.  Find the second derivative of f.  Set this to zero.  These are the points of inflection.

f = 3x^4+x^3-21x^2
f' = 12x^3+3x^2-42x
f'' = 36x^2+6x-42

Let me know if you have any specific questions.  Good luck.

Bobby