Answer:
y = 1/8x +1/64
Explanation:
You want the equation of the line that is tangent to the curve y = x³ -x⁴ in exactly two places.
Points of intersection
Since the tangent line is not crossed, the difference between the line and the given curve is a 4th-degree polynomial with two roots, each of multiplicity 2. If those are 'a' and 'b', we can write for line y = mx +p, ...
mx +p -(x³ -x⁴) = (x -a)²(x -b)² = 0
x⁴ -x³ +0x² +mx +p = x⁴ -2(a+b)x³ +(a²+4ab+b²)x² -2ab(a+b)x +a²b²
Equating coefficients, we have ...
- x⁴: 1 = 1
- x³: -1 = -2(a+b)
- x²: 0 = (a+b)² +2ab
- x: m = -2ab(a+b)
- constant: p = a²b²
Solution
The coefficient of x³ tells us ...
a +b = 1/2
The coefficient of x² tells us ...
0 = (a +b)² +2ab
0 = (1/2)² +2ab
ab = -1/8
Line
Using these value, we can find the equation of the line to be ...
m = -2ab(a+b) = -2(-1/8)(1/2) = 1/8
p = (ab)² = (-1/8)² = 1/64
The equation of the line is ...
y = 1/8x +1/64
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Additional comment
To find the line, we did not need the coordinates of the points of tangency. They are ((1-√3)/4, (3-2√3)/64) and ((1+√3)/4, (3+2√3)/64).
The slope of the line is equal to the slope of the quartic at the point of inflection of its derivative, at x=1/4.