I F up on my quiz, please show me how to do it. I don't get it

Question

Answer 1

(1)(a).

The function f(x) = -2x² + 3x + 3.

The derivative of f(x) = - 4x + 3.

Answer 2

(1).(b).

The function f(x) = -2x² + 3x + 3 and x = 2.

To find y-value, substitute x = 2 in the above equation.

f(x) = -2(2)² + 3(2) + 3 = - 8 + 6 + 3 = 1

The point is (2, 1).

Differentiate with respect to x to the original function.

f'(x) = -4x + 3

To find the slope of tangent line, substitute x = 2 in the above equation.

m = -4(2) + 3 = - 5.

Substitute (x₁, y₁) = (2, 1) and m = -5 in the point-slope form of line equation : y - y₁ = m(x - x₁).

y - (1) = (-5)(x - 2)

y - 1 = - 5x + 10

y = - 5x + 11

The tangent line equation is y = -5x + 11.

Answer 3

(2)(a).

The function is f(x) = x³ - 12x and the tangent line is horizontal line.

Therefore, the tangent line is y = b and its slope is zero.

Differentiate with respect to x to the original function.

f'(x) = 3x² - 12

Therefore, 3x² - 12 = 0.

x² - 4 = 0

x = 2 and x = -2.

If x = 2 then f(2) = (2)³ - 12(2) = 8 - 24 = - 16.

If x = - 2 then f(-2) = (-2)³ - 12(-2) = - 8 + 24 = 16.

The points are (2, -16) and (-2, 16).

Answer 4

(2)(b).

The function is f(x) = x³ - 12x and the slop of tangent line is 15.

Differentiate with respect to x to the original function.

f'(x) = 3x² - 12

Therefore, 3x² - 12 = 15.

3x² = 27

x² = 9

x = 3 and x = -3.

If x = 3 then f(3) = (3)³ - 12(3) = 27 - 36 = - 9.

If x = -3 then f(-3) = (-3)³ - 12(-3) = -27 + 36 = 9.

The points are (3, -9) and (-3, 9).

Answer 5

(2)(c).

The function is f(x) = x³ - 12x and the slop of tangent line is 36.

Differentiate with respect to x to the original function.

f'(x) = 3x² - 12

Therefore, 3x² - 12 = 36.

3x² = 48

x² = 16

x = 4 and x = -4.

If x = 4 then f(4) = (4)³ - 12(4) = 64 - 48 = 16.

If x = -4 then f(-4) = (-4)³ - 12(-4) = - 64 + 48 = - 16.

The points are (4, 16) and (-4, -16).

Answer 6

(3).

The derivative expression is (d/dx)(3x⁵ - 2x^3/2 - 7/x⁵ + √93).

= (d/dx)(3x⁵) - 2(d/dx)(x^3/2) - 7(d/dx)(1/x⁵) + (d/dx)(√93)

= 15x⁴ - 3x^1/2 + 35/x⁶

(d/dx)(3x⁵ - 2x^3/2 - 7/x⁵ + √93) = 15x⁴ - 3x^1/2 + 35/x⁶.

Answer 7

(4).

The solution is -2/x³.