Cubic polynomials

• Nov 17th 2009, 05:16 PM
Rumor
Cubic polynomials
Eh. Here's the problem:

"A cubic polynomial is given by \$\displaystyle f(x)=ax^3+bx^2+30x+k\$ where a, b, and k are constants. The function f(x) has a local minimum at (-1,-10) and a point of inflection at x=2.

a) Find the values of a, b, and k.

b) Use the second derivative to verify that (-1,-10) really is a local minimum for your function.

c) Find the location of any local maximums for your function. Verify that they are indeed a local maximum."

Here's what I've done so far..

I've found the first derivative, \$\displaystyle f(x)= 3ax^2 + 2bx + 30\$
and the second derivative, \$\displaystyle f(x)= 6ax + 2b\$.

I then plugged the value of x=1 into the first derivative, getting \$\displaystyle 3a-2b+30\$.
I also plugged the inflection point of x=2 into the second derivative and got
\$\displaystyle 12a+2b\$.

From there, I can't seem to remember what to do. Help, please? D:
• Nov 18th 2009, 05:22 AM
Shanks
solve the following linear equations :
\$\displaystyle f(-1)=a(-1)^3+b(-1)^2+30(-1)+k=-10\$
\$\displaystyle f'(-1)= 3a(-1)^2 + 2b(-1) + 30=0\$
\$\displaystyle f''(2)= 12a + 2b=0\$
to get the value of a, b, and k.