Question

Найти интервалы монотонности функции: y=3^4-8x^3-18x^2+1 Поподробнее пжлст

Answer 1

Y`=-24x²-36x=-12x(2x+3)=0
x=0  x=-1,5
               _                    +                        _
--------------------(-1,5)------------(0)----------------------
убыв              min     возр      max   убыв

Answer 2

Finding the Intervals of Monotonicity for the Function y = 3^4 - 8x^3 - 18x^2 + 1

To find the intervals of monotonicity for the function y = 3^4 - 8x^3 - 18x^2 + 1, we need to determine where the function is increasing or decreasing.

To do this, we can analyze the sign of the derivative of the function. If the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing.

Let's find the derivative of the function first:

y = 3^4 - 8x^3 - 18x^2 + 1

Taking the derivative with respect to x:

y' = 0 - 3 * 8x^2 - 2 * 18x

Simplifying:

y' = -24x^2 - 36x

Now, we need to find the critical points of the function by setting the derivative equal to zero and solving for x:

-24x^2 - 36x = 0

Factoring out -12x:

-12x(x + 3) = 0

Setting each factor equal to zero:

-12x = 0 or x + 3 = 0

Solving for x:

x = 0 or x = -3

Now, we have the critical points x = 0 and x = -3. We can use these critical points to determine the intervals of monotonicity.

To do this, we can choose test points within each interval and evaluate the sign of the derivative at those points. If the sign is positive, the function is increasing, and if the sign is negative, the function is decreasing.

Let's choose test points within the intervals:

1. Interval (-∞, -3): - Choosing x = -4, we can evaluate the sign of the derivative: y' = -24(-4)^2 - 36(-4) = -384 + 144 = -240 The sign of the derivative is negative, so the function is decreasing in this interval.

2. Interval (-3, 0): - Choosing x = -2, we can evaluate the sign of the derivative: y' = -24(-2)^2 - 36(-2) = -96 + 72 = -24 The sign of the derivative is negative, so the function is decreasing in this interval.

3. Interval (0, +∞): - Choosing x = 1, we can evaluate the sign of the derivative: y' = -24(1)^2 - 36(1) = -24 - 36 = -60 The sign of the derivative is negative, so the function is decreasing in this interval.

Based on the analysis above, we can conclude that the function y = 3^4 - 8x^3 - 18x^2 + 1 is decreasing in the intervals (-∞, -3), (-3, 0), and (0, +∞).

Please note that the intervals are inclusive of the endpoints if the derivative is defined at those points.

I hope this helps! Let me know if you have any further questions.