Question

This was a multiple choice question asking which of the following is an equation of the line tangent to the graph of f(x)=x^4+2x^2 at the point where f'(x)=1?

and the answer is y=x-0.122, but I don't understand why.

I tried to take the derivative of f(x) and got 4x^3+4x = 1 and solved for x, but I am not getting the answer. What am I doing wrong and how do I solve this?
Thanks!

you did nothing wrong, you are just faced with an equation very hard to solve.

4x^3 + 4x - 1 =0

I got x = .2367329 by a method called Newton's Method, which is sort of hard to explain in this forum.
substituting back in the original I go y=.1152257

so we have the point (.2367,.1152) and slope of 1
When used in y=mx+b I indeed got

y = x - .1215, they rounded it off to 3 decimals.

I don't know if you are in highschool or college, so I don't know what kind of tools you have to solve equations of that type.

how did you get b=.1215 though?

y=mx+b
.2367 = .1152 + b
b=.2367-.1152
b=.1215


The equation is not difficult to solve at all! You only need to know is a simple trick:

4x^3+4x - 1 = 0 --->

x^3 = -x + 1/4 (1)

The trick is to compare this with the equation for (a+b)^3, which you can write as:

(a+b)^3 = 3 a b(a+b) + a^3 + b^3 (2)

If you compare (2) to (1), you see that if you can find values for a and b such that

3 a b = -1 and a^3 + b^3 = 1/4 (3)

then a + b is a solution to the equation! But finding the numbers a and b such that (3) is satisfied amounts to solving a quadratic equation:

3 a b = -1 --->

a^3 b^3 = -1/27

So, if we put A = a^3 and B = b^3, then we have:

A B = -1/27

A + B = 1/4

A solution is:

A = 1/8 + sqrt[1/64 + 1/27]

B = 1/8 - sqrt[1/64 + 1/27]

And a solution for x is:

x = a + b = A^(1/3) + B^(1/3) =

[sqrt(1/64 + 1/27) + 1/8]^(1/3) -
[sqrt(1/64 + 1/27) - 1/8]^(1/3)

which gives the same result as Reiny gave above.

Answer 1

qckfu zoasubw ceyriow abke nyod ixkyj zqurci

Answer 2

(1/64)^x=Sqrt1/8