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Find the minimum and maximum of -4x^3-5x^2+10x-2

User Terrace
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Answer:

To find the minimum and maximum of the function -4x^3 - 5x^2 + 10x - 2, we can follow these steps:

1. Take the derivative of the function to find critical points:

f(x) = -4x^3 - 5x^2 + 10x - 2

f'(x) = -12x^2 - 10x + 10

2. Set the derivative equal to zero and solve for x to find the critical points:

-12x^2 - 10x + 10 = 0

3. Solve the quadratic equation by factoring or using the quadratic formula. In this case, factoring is not possible, so let's use the quadratic formula:

x = (-(-10) ± √((-10)^2 - 4(-12)(10))) / (2(-12))

Simplifying the expression inside the square root:

x = (-(-10) ± √(100 + 480)) / (-24)

x = (10 ± √580) / (-24)

4. Calculate the values of x by using the positive and negative solutions from the quadratic formula:

x ≈ 0.5486 or x ≈ -1.9653

5. Now, we need to determine whether these critical points correspond to a minimum or maximum. To do this, we can use the second derivative test. Take the second derivative of the function:

f''(x) = -24x - 10

6. Substitute the critical points into the second derivative:

For x ≈ 0.5486:

f''(0.5486) = -24(0.5486) - 10 ≈ -22.367

Since the second derivative is negative (-22.367 < 0), this indicates a maximum at x ≈ 0.5486.

For x ≈ -1.9653:

f''(-1.9653) = -24(-1.9653) - 10 ≈ 38.767

Since the second derivative is positive (38.767 > 0), this indicates a minimum at x ≈ -1.9653.

7. So, the minimum occurs at x ≈ -1.9653, and the maximum occurs at x ≈ 0.5486.

The minimum value of the function is obtained by substituting x ≈ -1.9653 into the original function:

f(-1.9653) = -4(-1.9653)^3 - 5(-1.9653)^2 + 10(-1.9653) - 2 ≈ -22.318

The maximum value of the function is obtained by substituting x ≈ 0.5486 into the original function:

f(0.5486) = -4(0.5486)^3 - 5(0.5486)^2 + 10(0.5486) - 2 ≈ 1.977

Therefore, the minimum value is approximately -22.318, and the maximum value is approximately 1.977.

User Arturs Vancans
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