Question

Найдите длину интервала убывания функции y=2x^3-3x^2-12x+5 Напишите решение,пожалуйста,на листочке

Answer 1

Никаких листочков, только хардкор (моё решение может быть неправильным, потому что я тупой)!
y'=6x^2-6x-12/6
y'=x^2-x-6
y'=0
x^2-x-6=0
D=3
x12=1+-3/2=2;-1
Луч x:
-+--[-1]---_---[2]---+-->x
От [-1] до [2] (если они входят, конечно).

Answer 2

Finding the Interval of Decrease for the Function y=2x^3-3x^2-12x+5

To find the interval of decrease for the function y=2x^3-3x^2-12x+5, we need to determine the values of x for which the function is decreasing. The interval of decrease will be the range of x-values over which the function's derivative is negative.

The first step is to find the derivative of the function y=2x^3-3x^2-12x+5. The derivative will give us information about the slope of the function at different points.

The derivative of the function y=2x^3-3x^2-12x+5 is given by: y' = 6x^2 - 6x - 12

Now, to find the interval of decrease, we need to solve the inequality y' < 0 for x.

Solving the Inequality for the Interval of Decrease

To solve the inequality 6x^2 - 6x - 12 < 0, we can use various methods such as factoring, the quadratic formula, or graphing. Let's use the quadratic formula to find the roots of the quadratic equation 6x^2 - 6x - 12 = 0.

The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a)

Where a = 6, b = -6, and c = -12.

Using the quadratic formula, we find the roots of the quadratic equation: x = (-(-6) ± √((-6)^2 - 4*6*(-12))) / (2*6) x = (6 ± √(36 + 288)) / 12 x = (6 ± √324) / 12 x = (6 ± 18) / 12

This gives us two possible values for x: x1 = (6 + 18) / 12 = 24 / 12 = 2 x2 = (6 - 18) / 12 = -12 / 12 = -1

So, the roots of the quadratic equation are x = 2 and x = -1.

Conclusion

The interval of decrease for the function y=2x^3-3x^2-12x+5 is given by the range of x-values where the derivative y' = 6x^2 - 6x - 12 is negative. From the solutions of the inequality 6x^2 - 6x - 12 < 0, we find that the function is decreasing for the interval -1 < x < 2.