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.
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