Question

2х³-9х²+12х-9 график функции

Answer 1

Ответ:

график на фото

область определения отсутствует

корень(3,0)

пересечение с осью y (0,-9)

Answer 2

To sketch the graph of the function $f(x) = 2x^3 - 9x^2 + 12x - 9$, we'll follow a step-by-step process:

Step 1: Find the x-intercepts (where the graph crosses the x-axis): To find the x-intercepts, set $f(x)$ to zero and solve for $x$: $2x^3 - 9x^2 + 12x - 9 = 0$

We can use factoring or other methods to find the roots, but in this case, we'll solve it numerically. Using a graphing calculator or software, we find the x-intercepts as follows: $x \approx -0.58$, $x \approx 2.79$, and $x \approx 3.79$.

Step 2: Find the y-intercept (where the graph crosses the y-axis): To find the y-intercept, set $x = 0$ in the equation $f(x) = 2x^3 - 9x^2 + 12x - 9$: $f(0) = 2(0)^3 - 9(0)^2 + 12(0) - 9 = -9$

So, the y-intercept is at the point (0, -9).

Step 3: Find critical points and inflection points: To find critical points, we need to find the values of $x$ for which the derivative of $f(x)$ is zero or undefined. Let's first find the derivative of $f(x)$: $f'(x) = d/dx(2x^3 - 9x^2 + 12x - 9) = 6x^2 - 18x + 12$

Next, set $f'(x) = 0$ and solve for $x$: $6x^2 - 18x + 12 = 0$

Divide through by 6: $x^2 - 3x + 2 = 0$

Factor the quadratic equation: $(x - 1)(x - 2) = 0$

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

Step 4: Find the concavity and inflection points: To find the concavity and inflection points, we need to find the second derivative of $f(x)$: $f''(x) = d^2/dx^2(2x^3 - 9x^2 + 12x - 9) = 12x - 18$

Set $f''(x) = 0$ and solve for $x$: $12x - 18 = 0$

Divide through by 6: $2x - 3 = 0$

$x = \frac{3}{2}$

Since $f''(x)$ is a linear function, there is only one inflection point, which is at $x = \frac{3}{2}$.

Step 5: Plot the points and sketch the graph: Now that we have all the critical points, x-intercepts, y-intercept, and inflection point, we can sketch the graph.

• The y-intercept is at (0, -9).
• The x-intercepts are approximately at (-0.58, 0), (2.79, 0), and (3.79, 0).
• The critical points are at (1, -4) and (2, -1).
• The inflection point is at ($\frac{3}{2}$, -$\frac{25}{2}$).

To understand the overall behavior of the function, we can also check the end behaviors. As $x$ approaches negative or positive infinity, the dominant term will be $2x^3$. Since the leading coefficient is positive, the graph will rise on both ends.

Combining all this information, the graph will look like:

yamlCopy code
          |       +           +       +            +
          |       |           |       |            |
          |       |           |       |            |
          |       |           |       |            |
    -9----+-------+-----------+-------+------------+-----> x
          |       |           |       |            |
          |       |           |       |            |
          |       |           |       |            |
          |       - - - - - - - - - - - - - - - -  |
          |                                        |

Please note that the sketch provided here is just a rough representation of the graph based on the critical points, intercepts, and end behavior. For a more accurate and detailed graph, you can use graphing software or calculators.