The Curve y=ax^3+bx^2+cx+5 passes through the point P (-1,4) and at the point P the first and second derivatives of the curve are 8 and -24 respectively. Find the values of the constants a,b and c?

1 Answer
Feb 6, 2018

(a,b,c)=(5,3,-1)

Explanation:

Let f be a real valued function defined by:
f(x)=ax^3+bx^2+cx+5

We have 3 conditions for the function:

  • f(-1)=4
  • f'(-1)=8
  • f''(-1)=-24

Start by computing the first and second derivatives:

f'(x)=3ax^2+2bx+c
f''(x)=6ax+2b

Substitute the function and it's derivatives in the condition and solve the simultaneous equations:

{a(-1)^3+b(-1)^2+c(-1)+5=4
{3a(-1)^2+2b(-1)+c=8
{6a(-1)+2b=-24

{c=1-a+b
{3a-2b+c=8
{b=3a-12

{c=-11+2a
{-3a+c=-16
{b=3a-12

{c=-11+2a
{a=5
{b=3a-12

{c=-1
{a=5
{b=3