Answer:
If I calculated correctly, the tangent line is horizontal where x ≈ -5.3 + 9.3i, and -5.3 - 9.3i
I'm somewhat concerned at having gotten complex numbers, and strongly recommend going through the steps to see if I missed anything. I checked it myself and don't see any errors.
Step-by-step explanation:
You can do this by taking the derivative of the function and solving for zero:
f(x) = 2x³ + 32x² + 220x + 11
f'(x) = 6x² + 64x + 220
f'(x) = 2(3x² + 32x + 110)
We can't factor that further, so let's do it the long way, starting by letting f'(x) equal zero:
0 = 2(3x² + 32x + 110)
0 = 3x² + 32x + 110
0 = 9x² + 96x + 990
0 = 9x² + 96x + 256 + 734
0 = (3x + 16)² + 734
(3x + 16)² = -734
3x + 16 = ± i√734
3x = -16 ± i√734
x = (-16 ± i√734) / 3
x ≈ (-16 + 27.9i) / 3, and (16 - 27.9i) / 3
x ≈ -5.3 + 9.3i, and -5.3 - 9.3i
I'm always wary when I end up with complex numbers. I'd suggest double checking everything here, but I'm fairly certain I did everything correctly.
