Quiz Turunan Fungsi

Explanation
The given function is f(x) = 3x^2 + 2x. To find the derivative of this function, we use the power rule for derivatives. The power rule states that the derivative of x^n is n*x^(n-1). Applying this rule, the derivative of 3x^2 is 2*3x^(2-1) = 6x. Similarly, the derivative of 2x is 2. Therefore, the derivative of f(x) = 3x^2 + 2x is f'(x) = 6x + 2.

Explanation
To find f'(x), we need to differentiate f(x) with respect to x. Taking the derivative of each term, the constant term 5 becomes 0, the coefficient of x (2) becomes 1, and the coefficient of x^2 (-3) becomes -6x. Therefore, f'(x) = 1 - 6x. To find f'(2), we substitute x=2 into the equation: f'(2) = 1 - 6(2) = 1 - 12 = -11. Hence, the correct answer is -11.

Explanation
To find the value of 6.f'(1) + f'(-1), we first need to find the derivative of f(x). Using the chain rule, we can differentiate each term separately. The derivative of (1 + 4x)^2 is 2(1 + 4x)(4), and the derivative of (2 - x) is -1. Multiplying these derivatives with the respective coefficients, we get 8(1 + 4x) and -2(1 + 4x). Evaluating these derivatives at x = 1 and x = -1, we get 40 and -10 respectively. Adding these values together, we get 40 - 10 = 30. Therefore, the answer is 10.

Explanation
The equation of the tangent line to the curve y = 2x^2 + 3x at the point (-2, 2) can be found by taking the derivative of the curve and evaluating it at x = -2. The derivative of y = 2x^2 + 3x is y' = 4x + 3. Plugging in x = -2, we get y' = 4(-2) + 3 = -8 + 3 = -5. So, the slope of the tangent line is -5. Using the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is the given point, we can substitute (-2, 2) and -5 into the equation to get y - 2 = -5(x - (-2)), which simplifies to y = -5x - 8. Therefore, the correct answer is y = -5x - 8.

Explanation
The given function is a cubic function, and the graph of a cubic function can either be increasing or decreasing. In this case, the graph of the function y = x^3 - 6x^2 + 9x + 2 is decreasing on the interval 1 < x < 3. This means that as x increases within this interval, the corresponding y-values decrease. Therefore, the correct answer is 1 < x < 3.

Explanation
The maximum value of a cubic function occurs at its vertex. To find the vertex, we can use the formula x = -b/2a, where a, b, and c are the coefficients of the function. In this case, the coefficient of x^3 is 1, the coefficient of x is -12, and the constant term is 1. Plugging these values into the formula, we get x = -(-12)/(2*1) = 6. Substituting this value back into the function, we get y = 6^3 - 12*6 + 1 = 217. Therefore, the maximum value of y is 217, not 17.

Explanation
The minimum value of a cubic function occurs at the point where its derivative is equal to zero. By finding the derivative of the given function, which is y = x^3 + 6x^2 - 15x - 2, and setting it equal to zero, we can solve for x. After solving, we find that x = -2. Plugging this value back into the original function, we get y = -10. Therefore, the minimum value of y is -10.

Explanation
The maximum profit that the company can obtain can be calculated by subtracting the production cost from the selling price multiplied by the number of gallons produced. In this case, the selling price is Rp. 400,000 per gallon and the production cost is given by the expression (x3 + 100x + 1500) thousand rupiah. Therefore, the profit function can be written as 400,000x - (x3 + 100x + 1500). To find the maximum profit, we need to find the value of x that maximizes this function. By differentiating the profit function with respect to x and setting it equal to zero, we can find the critical points. After analyzing the critical points, it is determined that the maximum profit is Rp. 500,000.