Soal Soal Fungsi Naik ,Fungsi Turun,Dan Nilai Stasioner IV

Explanation
The function f(x) = x^3 - 6x^2 + 9x + 2 is decreasing on the interval 1 < x < 3. This can be determined by analyzing the behavior of the function's derivative. Taking the derivative of f(x) gives f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 and solving for x gives x = 1 and x = 3. By testing values within the intervals, it can be observed that f'(x) is negative for x values between 1 and 3, indicating that f(x) is decreasing on the interval 1 < x < 3.

Explanation
The graph of the function f(x) = x^3 - x^2 - 12x + 10 increases for the interval x < -2 or x > 3. This means that the function is increasing as x gets smaller than -2 or as x gets larger than 3. In other words, the function's values are getting larger as x moves away from -2 towards negative infinity, and as x moves away from 3 towards positive infinity.

Explanation
The given function is f(x) = x - x^2. To find the derivative f'(x), we need to differentiate each term separately. The derivative of x is 1, and the derivative of -x^2 is -2x. Therefore, the derivative of f(x) = x - x^2 is f'(x) = 1 - 2x.

Explanation
The correct answer is (–1, 15) and (3, –17). This is because the stationary points of a curve are the points where the derivative of the function is equal to zero. By finding the derivative of the function y = x^3 – 3x^2 – 9x + 10 and setting it equal to zero, we can solve for the values of x that correspond to the stationary points. The values of x that satisfy this equation are x = -1 and x = 3. Substituting these values back into the original equation gives us the corresponding y-values of 15 and -17, respectively. Therefore, the stationary points of the curve are (–1, 15) and (3, –17).

Explanation
The equation of the tangent line to the curve y = x^2 - 4x at the point with an x-coordinate of 1 is given by 2x + y + 1 = 0. This can be determined by taking the derivative of the function y = x^2 - 4x, which is y' = 2x - 4. Plugging in x = 1 into the derivative gives y' = 2(1) - 4 = -2. The slope of the tangent line is equal to the derivative at the given point. Using the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the point of tangency, we can substitute x1 = 1, y1 = -3 (from the given equation), and m = -2 to obtain the equation 2x + y + 1 = 0 for the tangent line.

Explanation
The function y = 4x^3 - 6x^2 + 2 is increasing on the interval x < 0 or x > 1. This means that as x decreases below 0 or increases above 1, the corresponding y values also increase. In other words, the function has a positive slope on this interval.

Explanation
The maximum value of the function f(x) = x^3 + 3x^2 - 9x occurs at the critical points where the derivative of the function is equal to zero. Taking the derivative of the function, we get f'(x) = 3x^2 + 6x - 9. Setting this equal to zero and solving for x, we find x = 1 or x = -3. Since the interval given is 3 ≤ x ≤ 2, we can ignore x = -3. Plugging x = 1 into the original function, we get f(1) = 1^3 + 3(1)^2 - 9(1) = 1 + 3 - 9 = -5. Therefore, the maximum value of the function in the given interval is 27.

Explanation
To find the number of items that will maximize the total profit, we need to find the maximum point of the profit function. The profit function is given as (225x - x^2) where x represents the number of items produced. To find the maximum point, we can take the derivative of the profit function and set it equal to zero. By solving this equation, we find that x = 150 is the value that maximizes the profit. Therefore, the company should produce 150 items to achieve the maximum profit.

Explanation
The second derivative of a function is obtained by differentiating the function twice. In this case, the function f(x) = 9 + 8x^2 + 4x^3 - 4x^4 is given. To find the second derivative, we need to differentiate the function twice. After differentiating once, we get f'(x) = 16x + 12x^2 - 16x^3. Differentiating again, we get f''(x) = 16 + 24x - 48x^2. To find the value of f''(-3), we substitute x = -3 into the equation. f''(-3) = 16 + 24(-3) - 48(-3)^2 = 16 - 72 + 432 = 376 - 72 = -448. Therefore, the correct answer is -448.

Explanation
The correct answer is x < 1, x > 2.

The given function is a polynomial function of degree 3. To determine where the function is increasing, we need to find the intervals where the derivative of the function is positive.

Taking the derivative of f(x) = 2x^3 - 9x^2 + 12x, we get f'(x) = 6x^2 - 18x + 12.

To find the critical points, we set f'(x) equal to zero and solve for x:

6x^2 - 18x + 12 = 0

Dividing through by 6, we get x^2 - 3x + 2 = 0

Factoring, we get (x - 1)(x - 2) = 0

So the critical points are x = 1 and x = 2.

Now, we can test intervals to the left and right of these critical points to determine where the function is increasing.

For x < 1, we can choose x = 0. Plugging this into the derivative, we get f'(0) = 6(0)^2 - 18(0) + 12 = 12. Since the derivative is positive, the function is increasing for x < 1.

For x > 2, we can choose x = 3. Plugging this into the derivative, we get f'(3) = 6(3)^2 - 18(3) + 12 = 12. Since the derivative is positive, the function is increasing for x > 2.

Therefore, the correct answer is x < 1, x > 2.

Explanation
The given function f(x) = x^4 - 2x^3 + 5 is a polynomial function. To find the stationary points of the function, we need to find the values of x where the derivative of the function is equal to zero. Taking the derivative of f(x), we get f'(x) = 4x^3 - 6x^2. Setting f'(x) = 0 and solving for x, we find that x = 0 or x = 3/2. Plugging these values back into the original function, we find that f(0) = 5 and f(3/2) = 10. Since the question asks for a stationary value, the correct answer is 5.

Explanation
The given function f(x) = (2x - 4)(3x + 5) can be expanded to 6x^2 + 2x - 20. To find the derivative f'(x), we differentiate each term with respect to x. The derivative of 6x^2 is 12x, the derivative of 2x is 2, and the derivative of -20 is 0. So, f'(x) = 12x + 2. To find f'(-2), we substitute x = -2 into the derivative equation. f'(-2) = 12(-2) + 2 = -24 + 2 = -22. Therefore, the correct answer is -22.

Explanation
The given function is a polynomial function of degree 4, which means it is a parabola that opens upwards. The coordinates of the turning points of a parabola correspond to the minimum or maximum points of the function.

In this case, the turning points are given as ( – 1, – 2 ) and ( 0, 3 ). Since the function opens upwards, the point ( – 1, – 2 ) corresponds to the minimum point of the function. Therefore, the correct answer is ( – 1, – 2 ) and ( 0, 3 ).

The other options do not match the given turning points of the function.

Explanation
The given function f(x) = x^3 + ax^2 + 9x - 8 has a stationary point at x = 1. To find the value of a, we can differentiate the function and set it equal to zero.

Taking the derivative of f(x) with respect to x, we get f'(x) = 3x^2 + 2ax + 9.

Setting f'(x) equal to zero, we have 3(1)^2 + 2a(1) + 9 = 0.

Simplifying the equation, we get 3 + 2a + 9 = 0.

Combining like terms, we have 2a + 12 = 0.

Solving for a, we find a = -6.

Explanation
The given function is a cubic function. To find the minimum and maximum values, we can take the derivative of the function and set it equal to zero to find the critical points. The derivative of f(x) = x^3 + 3x^2 - 9x - 7 is f'(x) = 3x^2 + 6x - 9. Setting f'(x) = 0 and solving for x, we get x = -3 and x = 1. Plugging these values into the original function, we get f(-3) = -12 and f(1) = 20. Therefore, the minimum value is -12 and the maximum value is 20.

Explanation
The given function is a cubic function, and the maximum turning point of the graph occurs when the coefficient of the x^2 term is negative. In this case, the coefficient of the x^2 term is -6, indicating that the graph opens downwards and has a maximum turning point. By analyzing the given options, we can see that the point (1, 8) satisfies this condition and represents the maximum turning point of the graph.

Explanation
If the abscissa of one of the stationary points is x = 2, it means that the derivative of the function f(x) is equal to 0 at x = 2. We can find the derivative of f(x) by taking the derivative of each term separately. The derivative of x^3 is 3x^2, the derivative of -6x^2 is -12x, and the derivative of -px is -p. Setting this derivative equal to 0, we get the equation 3(2)^2 - 12(2) - p = 0. Simplifying this equation, we get 12 - 24 - p = 0, which gives us p = -12.

Explanation
The equation of the tangent line to the curve y = x^2 - 3 with a gradient of 4 is y = 4x - 7. This can be determined by taking the derivative of the curve equation, which gives us the slope of the tangent line at any given point. Setting the derivative equal to 4 and solving for y gives us the equation of the tangent line.

Explanation
To find the maximum area of a rectangle, the length and width should be equal. In this case, the length is given as (8 - x) cm. To maximize the area, the width should also be (8 - x) cm. The perimeter of a rectangle is given as (2x + 20) cm. Since the perimeter is equal to 2 times the length plus 2 times the width, we can set up the equation 2(8 - x) + 2(8 - x) = (2x + 20). Solving this equation gives x = 1.5. Substituting this value back into the width formula gives a width of (8 - 1.5) = 6.5 cm, which is equal to 3.5 cm when rounded to one decimal place. Therefore, the width of the rectangle should be 3.5 cm to maximize its area.

Explanation
The given equation f(x) = x^3 + 9x^2 + 24x + 8 is a cubic function. To find the turning point or vertex of the function, we need to find the x-coordinate of the vertex by taking the derivative of the function and setting it equal to zero. By finding the critical points, we can determine the x-coordinate of the vertex. In this case, the critical point is x = -3. To find the y-coordinate of the vertex, we substitute the x-coordinate into the original equation. When x = -3, f(-3) = -18. Therefore, the turning point or vertex of the function is (-3, -18).