The second derivative f''(x) = e^x.
f''(x) = 12x - 12.
Slope = (6-2)/(3-1) = 4/2 = 2.
Slope = (7-3)/(4-2) = 4/2 = 2.
The slope of the given line is -2. The slope of the perpendicular line is the negative reciprocal, which is 1/2.
The slope of the given line is -3. The slope of the perpendicular line is the negative reciprocal: 1/3.
Rearranging gives y = (2/3)x + 2, slope = 2/3.
Rearranging gives y = (3/4)x + 3. Slope = 3/4.
Rearranging to slope-intercept form: 5y = 10x + 20 => y = 2x + 4. The slope is 2.
The slope m = (8 - 0) / (4 - 0) = 2.
The equation can be factored to find the slopes, which are both 1.
Factoring gives (x - 3y)^2 = 0, indicating a double root, hence the slope is 3.
The derivative y' = 2x + 2. At x = 1, y' = 2(1) + 2 = 4, which is the slope of the tangent line.
4 - x < 2 => -x < -2 => x > 2.
4x + 1 < 3x + 5 => x < 4.
4x - 1 < 3x + 2 => x < 3.
6 - 2x ≥ 0 => -2x ≥ -6 => x ≤ 3.
7 - 4x < 3 => -4x < -4 => x > 1.
7x + 2 ≤ 5x + 10 => 2x ≤ 8 => x ≤ 4.
7x + 3 < 24 => 7x < 21 => x < 3.
8 - 3x > 2 => -3x > -6 => x < 2.
2x + 5 > 3x - 1 => 6 > x => x < 6.
This is a separable equation. Integrating gives y = Ce^(x^3/3 + x).
The characteristic equation has a repeated root, leading to the solution form (C1 + C2x)e^(2x).
-x + 6 > 0 => -x > -6 => x < 6.
4x + 1 < 9 => 4x < 8 => x < 2.
5 - 2x < 1 => -2x < -4 => x > 2.
6x - 4 < 2x + 8 => 4x < 12 => x < 3.
