The function is a downward-opening parabola. The maximum occurs at x = -b/(2a) = -8/(2*-2) = 2. f(2) = -2(2^2) + 8(2) - 3 = 8.

Midpoint = ((2+4)/2, (3+7)/2) = (3, 5).

The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4*2 + 7 = 3.

The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4*2 + 7 = 3.

The minimum occurs at x = 2. f(2) = 2(2^2) - 8(2) + 10 = 6.

The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).

The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).

f''(x) = 6x - 12. Setting f''(x) = 0 gives x = 2. f(2) = 3.

To find horizontal tangents, set the derivative y' = 3x^2 - 6x = 0. This gives x = 0 and x = 2. The point (1, 2) has a horizontal tangent.

The real part of the complex number 4 + 5i is 4.

Factoring gives (x + 4)(x - 2) = 0, hence the roots are 4 and -2.

f'(x) = 6x^2 - 6. f'(1) = 6(1)^2 - 6 = 0.

This is a separable equation. Integrating gives y = 1/(C - x).

This is a linear first-order equation. The integrating factor is e^(3x). The solution is y = Ce^(3x) + 2.

This is a separable equation. The solution is y = tan(x + C).

This is a first-order linear differential equation. The solution is y = Ce^(-2x).

The term containing x^3 is C(5,3) * x^3 * (-1)^2 = 10 * x^3 * 1 = 10.

The term independent of x occurs when the powers of x cancel out. The term is 81.

The term independent of x occurs when the powers of x cancel out. The term is C(5,2)(-3)^2(1)^3 = 45.

The expression can be rewritten as (x - 2)^4. The term independent of x occurs when k = 4, which gives us (-2)^4 = 16.

Using the binomial theorem, (3 + 2)^3 = C(3,0) * 3^3 * 2^0 + C(3,1) * 3^2 * 2^1 + C(3,2) * 3^1 * 2^2 + C(3,3) * 3^0 * 2^3 = 27 + 54 + 36 + 8 = 125.

The discriminant must be negative: 4² - 4*1*k < 0, which gives k > 4, so the minimum value is -6.

For equal roots, the discriminant must be zero: k² - 4*1*16 = 0, thus k² = 64, k = ±8. The value of k can be -8.

For no real roots, the discriminant must be less than zero: k² - 4*1*16 < 0, which gives k > 8.

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

Setting x = 0 in the equation gives y = 8. Therefore, the y-intercept is 8.

The integral is (5/5)x^5 + C = x^5 + C.

The integral is (6/3)x^3 - 4x + C = 2x^3 - 4x + C.

The integral of e^x is e^x + C.

The discriminant must be non-negative: 6² - 4*1*k ≥ 0, which gives k ≤ 9, so the minimum value is -9.