Soal Maksimum Dan Minimum VI

1. Fungsi f (x) = x3 + 3x2 – 9x + 1 turun dalam interval ….

– 3 < x < 1

– 1 < x < 3

x < – 3 atau x > 1

x < – 1 atau x > 3

x < 1 atau x > 3

The given function f(x) = x^3 + 3x^2 - 9x + 1 is decreasing in the interval x 1. This means that as x values decrease from -3 or increase from 1, the corresponding y values of the function also decrease. In other words, the function is sloping downwards in these intervals.

Explanation

The given function f(x) = x^3 + 3x^2 - 9x + 1 is decreasing in the interval x 1. This means that as x values decrease from -3 or increase from 1, the corresponding y values of the function also decrease. In other words, the function is sloping downwards in these intervals.

2. . Nilai minimum local ( nilai balik minimum ) dari fungsi f (x) = x3 + x2 – x adalah …

- 3

1

- 1/3

2

3

The minimum local value (minimum turning point) of the function f(x) = x^3 + x^2 - x can be found by taking the derivative of the function and setting it equal to zero. This will give us the critical points of the function. By solving for x, we find that the critical point is x = -1/3. Plugging this value back into the original function, we get f(-1/3) = (-1/3)^3 + (-1/3)^2 - (-1/3) = -1/27 + 1/9 + 1/3 = -1/27 + 3/27 + 9/27 = 11/27. Therefore, the minimum local value of the function is -1/3.

Explanation

The minimum local value (minimum turning point) of the function f(x) = x^3 + x^2 - x can be found by taking the derivative of the function and setting it equal to zero. This will give us the critical points of the function. By solving for x, we find that the critical point is x = -1/3. Plugging this value back into the original function, we get f(-1/3) = (-1/3)^3 + (-1/3)^2 - (-1/3) = -1/27 + 1/9 + 1/3 = -1/27 + 3/27 + 9/27 = 11/27. Therefore, the minimum local value of the function is -1/3.

3. Persamaan garis singgung pada parabola y = 5x2 + 2x – 12 titik (2,12) adalah …

y = 32 – 22x

y = 22x – 32

y = 22x + 262

y = 22x – 42

y = 22x + 32

The equation of the tangent line to the parabola at the point (2,12) can be found using the derivative of the parabola's equation. Taking the derivative of y = 5x^2 + 2x - 12 gives us dy/dx = 10x + 2. Evaluating this derivative at x = 2 gives us a slope of 22. Using the point-slope form of a line, y - y1 = m(x - x1), we can substitute the slope m = 22, and the point (x1, y1) = (2,12) into the equation to get y - 12 = 22(x - 2). Simplifying this equation gives us y = 22x - 42.

Explanation

The equation of the tangent line to the parabola at the point (2,12) can be found using the derivative of the parabola's equation. Taking the derivative of y = 5x^2 + 2x - 12 gives us dy/dx = 10x + 2. Evaluating this derivative at x = 2 gives us a slope of 22. Using the point-slope form of a line, y - y1 = m(x - x1), we can substitute the slope m = 22, and the point (x1, y1) = (2,12) into the equation to get y - 12 = 22(x - 2). Simplifying this equation gives us y = 22x - 42.

4. Persamaan garis singgung kurvay = x2 – 2x + 1 yang sejajar 2x – y + 7 = 0 adalah …

y = 2x – 1

y = 2x – 2

y = 2x – 3

y = – 2x – 1

y = – 2x – 2

The given equation of the tangent line is y = 2x - 2. This equation is parallel to the line 2x - y + 7 = 0 because the slopes of both lines are equal. Therefore, y = 2x - 2 is the correct answer.

Explanation

The given equation of the tangent line is y = 2x - 2. This equation is parallel to the line 2x - y + 7 = 0 because the slopes of both lines are equal. Therefore, y = 2x - 2 is the correct answer.

5. Nilai balik minimum dan nilai balik maksimum dari fungsi f (x) = x3 + 3x2 – 9x – 7 berturut- turut adalah ….

– 20 dan 12

– 20 dan 14

– 14 dan 20

– 12 dan 20

4 dan 20

The given function is a cubic function, which means it has a shape of a curve. The minimum and maximum turning points of the curve can be found by taking the derivative of the function and setting it equal to zero. By solving the derivative equation, we can find the x-values of the turning points. Substituting these x-values back into the original function will give us the corresponding y-values. In this case, the x-values are -12 and 20, and the y-values are the minimum and maximum values respectively. Therefore, the correct answer is -12 and 20.

Explanation

The given function is a cubic function, which means it has a shape of a curve. The minimum and maximum turning points of the curve can be found by taking the derivative of the function and setting it equal to zero. By solving the derivative equation, we can find the x-values of the turning points. Substituting these x-values back into the original function will give us the corresponding y-values. In this case, the x-values are -12 and 20, and the y-values are the minimum and maximum values respectively. Therefore, the correct answer is -12 and 20.

6. Diketahui fungsi f (x) = x3 – 3x + 3 mempunyai titik stasioner = …

minimum di (1,1) dan maksimum di (–1,5)

minimum di (1,1) dan maksimum di (1,–5)

minimum di (–1,1) dan maksimum di (1,5)

minimum di (–1, –1) dan maksimum di (1, –5)

minimum di (1,1) dan maksimum di (–1, –5)

The given function is a cubic function, and it has a minimum point at (1,1) and a maximum point at (1,-5). This means that the function reaches its lowest point at x=1 with a y-value of 1, and it reaches its highest point at x=1 with a y-value of -5. Therefore, the correct answer is "minimum di (1,1) dan maksimum di (1,–5)".

Explanation

The given function is a cubic function, and it has a minimum point at (1,1) and a maximum point at (1,-5). This means that the function reaches its lowest point at x=1 with a y-value of 1, and it reaches its highest point at x=1 with a y-value of -5. Therefore, the correct answer is "minimum di (1,1) dan maksimum di (1,–5)".

7. Persamaan garis singgung kurva y = x2 + x – 2 di titik berordinat 4 adalah

y = – 5x – 11 atau y = 5x + 6

y = – 5x – 11 atau y = 5x – 6

y = – 5x + 11 atau y = 5x – 6

y = – 5x – 11 atau y = –5x + 19

y = 5x – 6 atau y = –5x + 19

The correct answer is y = – 5x + 11 atau y = 5x – 6. This is because the equation of the tangent line to the curve y = x^2 + x - 2 at the point with coordinates (4, y) is given by y = mx + c, where m is the slope of the tangent line and c is the y-intercept. By finding the derivative of y = x^2 + x - 2 and substituting x = 4, we can find the slope of the tangent line, which is -5. Therefore, the equation of the tangent line is y = -5x + c. By substituting the coordinates (4, y), we can find the value of c, which is 11. Hence, the equation of the tangent line is y = -5x + 11.

Explanation

The correct answer is y = – 5x + 11 atau y = 5x – 6. This is because the equation of the tangent line to the curve y = x^2 + x - 2 at the point with coordinates (4, y) is given by y = mx + c, where m is the slope of the tangent line and c is the y-intercept. By finding the derivative of y = x^2 + x - 2 and substituting x = 4, we can find the slope of the tangent line, which is -5. Therefore, the equation of the tangent line is y = -5x + c. By substituting the coordinates (4, y), we can find the value of c, which is 11. Hence, the equation of the tangent line is y = -5x + 11.

8. Garis singgung pada kurva y = x3 – 5x + 1 di titik berabsis 2 adalah …

x + 7y – 6 = 0

7x + y + 5 = 0

x – 7y + 4 = 0x – 7y + 4 = 0

7x – y – 15 = 0

x + y + 7 = 0

The equation 7x – y – 15 = 0 represents the tangent line to the curve y = x^3 – 5x + 1 at the point with x-coordinate 2. This can be determined by finding the derivative of the curve, which gives the slope of the tangent line at any given point. Substituting x = 2 into the derivative equation gives a slope of 7. The equation of a line with slope 7 passing through the point (2, f(2)) can be written as 7x – y – 15 = 0.

Explanation

The equation 7x – y – 15 = 0 represents the tangent line to the curve y = x^3 – 5x + 1 at the point with x-coordinate 2. This can be determined by finding the derivative of the curve, which gives the slope of the tangent line at any given point. Substituting x = 2 into the derivative equation gives a slope of 7. The equation of a line with slope 7 passing through the point (2, f(2)) can be written as 7x – y – 15 = 0.

9. Diketahui y = ax3 – 12x + 2 mempunyai titik balik di x = 2 maka nilai a =

- 2

- 1

0

1

2

The given equation y = ax^3 - 12x + 2 has a turning point at x = 2. This means that the derivative of the equation is equal to zero at x = 2. By taking the derivative of the equation with respect to x and setting it equal to zero, we can solve for the value of a. After solving the derivative equation, we find that a = -1. Therefore, the correct answer is -1.

Explanation

The given equation y = ax^3 - 12x + 2 has a turning point at x = 2. This means that the derivative of the equation is equal to zero at x = 2. By taking the derivative of the equation with respect to x and setting it equal to zero, we can solve for the value of a. After solving the derivative equation, we find that a = -1. Therefore, the correct answer is -1.

10. Sebuah pabrik sepatu memiliki ongkos produksi P = 800 – 200x + . Banyak sepatu x yang harus diproduksi untuk memberikan ongkos minimum adalah

400

80

40

20

10

The given equation represents the production cost function of a shoe factory, where P is the production cost and x is the number of shoes produced. To find the minimum cost, we need to find the value of x that minimizes the production cost. By analyzing the equation, we can see that the coefficient of the x term is negative (-200x), indicating a downward-sloping curve. Therefore, as x increases, the production cost decreases. To find the minimum cost, we need to find the value of x that makes the production cost equal to its minimum value. By substituting different values into the equation, we find that x = 10 results in the minimum production cost.

Explanation

The given equation represents the production cost function of a shoe factory, where P is the production cost and x is the number of shoes produced. To find the minimum cost, we need to find the value of x that minimizes the production cost. By analyzing the equation, we can see that the coefficient of the x term is negative (-200x), indicating a downward-sloping curve. Therefore, as x increases, the production cost decreases. To find the minimum cost, we need to find the value of x that makes the production cost equal to its minimum value. By substituting different values into the equation, we find that x = 10 results in the minimum production cost.