7x + 3 < 4x + 12 => 3x < 9 => x < 3.

Add 4 to both sides: 7x ≥ 14. Then divide by 7: x ≥ 2.

Setting f(1-) = f(1+) and f'(1-) = f'(1+) leads to a = 1 for differentiability.

Setting the two pieces equal at x = 1 gives us 3 + c = 1. Thus, c = -2.

To ensure continuity at x = 1, k(1) + 1 = 2(1) - 3, solving gives k = 2.

To ensure continuity at x = 1, we need to set the two pieces equal: 1^2 + k = 2(1) + 1. This gives k = 2.

To ensure continuity at x = 1, we need to set the two pieces equal: k + 1^2 = 2(1) + 3. This gives k + 1 = 5, so k = 4.

For f(x) to be continuous at x = 2, we need limit as x approaches 2 from left to equal limit as x approaches 2 from right and equal to f(2). Thus, k = 4.

For f(x) to be continuous at x = 2, we need limit as x approaches 2 from left to equal limit as x approaches 2 from right. Thus, k must equal 0.

Setting f(2-) = f(2+) and f'(2-) = f'(2+) leads to m = 1 for differentiability.

Setting the two pieces equal at x = 0: n^2 - 1 = 3. Solving gives n = ±2.

Setting 2(2) + 3 = p(2) + 1 gives p = 3 for continuity.

Setting 3(2) - 1 = p(2) + 4 gives p = 2.

Setting p = 1 for continuity at x = 0 gives f(0) = 1.

Setting the left limit (1 - 1 = 0) equal to the right limit (2(1) + 1 = 3), we find p = 2.

Setting 1 - 3 + p = 2 + 1 gives p = 4.

Setting -3 + p = 3 gives p = 0.

Setting the determinant to zero and solving gives \( k = 10 \).

The solutions are x = π/4 and x = 3π/4.

The solutions are x = 0, π/2, π, 3π/2.

The solutions are x = nπ/2, where n is any integer.

Factoring gives sin(x)(sin(x) - 1) = 0, so x = 0, π/2, π, 3π/2.

Setting y = 0 in the equation gives 4x + 8 = 0, thus x = -2.

Setting y = 0 in the equation gives 4x + 20 = 0, thus x = -5.

Setting y = 0 in the equation gives 5x - 10 = 0, thus x = 2.

Set y = 0 in the equation: 2x + 6 = 0 => x = -3.

Setting y = 0 in the equation gives 5x = 10, thus x = 2. The x-intercept is 2.

Using the product rule, f'(x) = 4e^x + 4x^2e^x.

Using the chain rule, f'(x) = 2x/(x^2 + 1).

Using the product rule, f'(x) = x^2 * e^x + 2x * e^x.