Question

Suppose the curve y=x4+ax3+bx2+cx+d has a tangent line when x=0 with equation y=2x+4 and a tangent line when x=1 with equation y=4x+1. Find the values of a, b, c, and d. Answer: a= b= c= d=

Answer 1

y'=4x^3 +3ax^2 +2bx +c

Answer 2

when x=0, y=d, then find the equation of the tangent line at (0,d) . You should get a function in terms of c,d. This function must be equal to y=2x+4
when x=1, y=1+a+b+c+d , then find the equation of the tangent line at (1,1+a+b+c+d)
.You should get a function in terms of a,b,c,d.This function should be equal to y=4x+1.

Answer 3

Firstly, we know that since y (4th degree function) is tangent to 2x + 4 at x = 0, y must have the same value as 2x + 4 at x = 0, hence at x = 0, y = 4. This implies that \(\bf\ d=4\) since all the other terms will cancel out at x = 0.

Now since these lines are tangent, then this means that their slopes are the derivatives of the y at the respective value of x. The derivative of y is:\[\bf y'=4x^3+3ax^2+2bx+c\]And we know that at x = 0, the tangent line has slope of 2 so y' must equal 2 at x = 0:\[\bf 4(0)^3+3a(0)^2+2b(0)+c=2 \implies c=2\] Now we need to solve for \(\bf a\) and \(\bf b\). Since we know the value of c, we can re-write our derivative as:\[\bf y'=4x^3+3ax^2+2bx+2\]We know our derivative is 4 at x = 1, so now can plug in x = 1 once again into our derivative to see what we get which can help us solve for a and b:\[\bf 4(1)^3+3a(1)^2+2b(1)+2=4 \implies 3a+2b=-2\]Hmmm, now if we could get a system of equations, we could solve for a and b. But how can we do that? Let's try plugging in x = 1 into our original function by including the known values of c and d:\[\bf (1)^4+a(1)^3+b(1)^2+2(1)+4=5 \implies a+b=-2\]Now we have a sytem of equations for a and b! Let's solve for a and b now:\[\bf 3a+2b=-2\]\[\bf a+b=-2 \implies 2a+2b=-4\]Let's eliminate b by subtracting the lower equation from the top: \[\bf 3a+2b-2a+2b=-2-(-4) \implies a=2\]Plugging a back in to one of the above equations will allow us to solve for b:\[\bf a + b=-2 \rightarrow 2+b=-2 \implies b = -4\]And we are done! Pretty straight-forward and easy if you get the thinking right.

\(\bf \therefore a=2,b=-4,c=2,d=4\)

@Invizen

Answer 4

Thank you so much! Such a professional way to answer might I add

Answer 5

No problem.