To find inflection points, set f''(x) = 0. f''(x) = 6x - 12. Setting 6x - 12 = 0 gives x = 2.

To find inflection points, set f''(x) = 0. f''(x) = 6x - 12. Setting 6x - 12 = 0 gives x = 2.

Subtracting 5 from both sides gives 3x = 15, thus x = 15/3 = 5.

Setting f'(x) = 3x^2 - 12x + 9 = 0 gives x = 1, 3. Testing shows x = 2 is a local minimum.

To find inflection points, set f''(x) = 0. f''(x) = 6x - 12. Setting it to zero gives x = 2.

Using the quadratic formula, z = [-4 ± √(16 - 32)]/2 = -2 ± 2i.

Taking square root, z = ±√(-16) = ±4i.

z^2 = (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i.

Setting the determinant to zero gives \( k = 6 \).

Setting the determinant to zero: \( 1*x - 2*3 = 0 \) gives \( x - 6 = 0 \) or \( x = 6 \).

The determinant of the identity matrix is always 1.

By definition, \( \sin(\sin^{-1}(x)) = x \). Therefore, \( \sin(\sin^{-1}(\frac{1}{2})) = \frac{1}{2} \).

The integral evaluates to [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.

The integral evaluates to [e^x] from 0 to 1 = e - 1.

Using integration by parts, the integral evaluates to (e - 1)/2.

The integral evaluates to [x^3/3 - 1/x] from 0 to 1 = (1/3 - 1) = -2/3.

The integral evaluates to [x^3/3 + (3/2)x^2 + 2x] from 0 to 1 = (1/3 + 3/2 + 2) - 0 = 4/3 + 3/2 = 2.5.

The integral evaluates to [x^3/3 - x^2 + x] from 0 to 1 = (1/3 - 1 + 1) = 1/3.

The integral evaluates to [x^4/4 - x^3 + 2x] from 0 to 1 = (1/4 - 1 + 2) - (0) = 1/4.

The integral evaluates to [x^4/4 - x^3 + (3/2)x^2 - x] from 0 to 1 = 0.

The integral evaluates to [x^4/4 - x^3 + (3/2)x^2] from 0 to 1 = (1/4 - 1 + 3/2) = 1/4.

The integral evaluates to [x^4/4 - 2x^2 + 4x] from 0 to 1 = (1/4 - 2 + 4) = (1/4 + 2) = 9/4.

The integral evaluates to [x^5/5 + (2/3)x^3] from 0 to 1 = (1/5 + 2/3) = 11/15.

The integral evaluates to [x^5/5 - x^4 + 2x^3 - 2x^2 + x] from 0 to 1 = (1/5 - 1 + 2 - 2 + 1) = 0.

The integral evaluates to [x^5/5] from 0 to 1 = 1/5.

The integral evaluates to [x^3/3 - x^2 + x] from 0 to 2 = (8/3 - 4 + 2) = 2/3.

The integral evaluates to [x^3 - 2x] from 1 to 2 = (8 - 4) - (1 - 2) = 4 + 1 = 5.

The integral evaluates to [x^3 - x^2 + x] from 1 to 2 = (8 - 4 + 2) - (1 - 1 + 1) = 5.

The integral evaluates to [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.

∫_0^1 (4x^3) dx = [x^4] from 0 to 1 = 1.