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#?

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Answer 1 · 2018-02-06T19:14:39

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

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

We have 3 conditions for the function:

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#