Question

Suppose this curve y=x^4+ax^3+bx^2+cx+d has a tangent line when x=0 with equation y=2x+1 and a tangent line when x=1 with equation y=6x−3. Find the values of a, b, c, and d.

Answer 1

First differentiate.

Answer 2

\(f'(0)=2\) and \(f'(1)=6\)

Answer 3

so c=0

Answer 4

No, \(c=2\).

Answer 5

if x=0 then everything cancels out except for c
4x^3+3ax^2+2bx+c

Answer 6

for tangent line:
y = mx+ b
\[m = f'(x)\]

Answer 7

Yeah and \(f'(0)=2\) so you end up with \(c=2\).

Answer 8

ok and a=(-2/3)b

Answer 9

Okay. \(f(0)=1\)

Answer 10

and \(f(1)=6(1)-3=3\)

Answer 11

Because we know tangent lines have to intersect at these points to be tangent.

Answer 12

ok so

Answer 13

set f(1) = 3 and solve for "b" given c=2, d=1, a = (-2/3)b

Answer 14

b=-3?

Answer 15

@dumbcow

Answer 16

yes

Answer 17

so would i do the same thing to find a

Answer 18

i got it. a=2

Answer 19

yep

Answer 20

thanks @dumbcow and @wio