Question

F(x) = x2 In(x - 10);​

Answer 1

Ответ:

$f(x)=x^2*ln(x-10)\\f'(x)=2x*ln(x-10)+x^2*\frac{1}{x-10}$

Объяснение:

Answer 2

To analyze the function $F(x) = x^2 \ln(x - 10)$, let's break down its properties and explore its behavior.

The function $F(x) = x^2 \ln(x - 10)$ is a combination of two functions: $x^2$ and $\ln(x - 10)$. The domain of this function depends on the individual domains of $x^2$ and $\ln(x - 10)$.

1. Domain:

   • $x^2$ is defined for all real numbers.
   • $\ln(x - 10)$ is defined for $x > 10$ since the natural logarithm of a non-positive number is not real.

   Therefore, the domain of the function $F(x)$ is $x > 10$.

2. X-intercept: The x-intercept is the point where the graph intersects the x-axis (i.e., $F(x) = 0$). $x^2 \ln(x - 10) = 0$ Since the natural logarithm of zero is undefined, there is no real x-intercept for this function.

3. Y-intercept: The y-intercept is the point where the graph intersects the y-axis (i.e., $x = 0$). $F(0) = 0^2 \ln(0 - 10) = 0$ The y-intercept is at the origin (0, 0).

4. Symmetry: The function $F(x)$ is not symmetric about the y-axis (even) or the origin (odd) since it contains both $x^2$ (even) and $\ln(x - 10)$ (neither even nor odd).

5. Asymptotes: The function $F(x)$ may have vertical asymptotes if $\ln(x - 10)$ approaches negative infinity for some $x$ in the domain. However, this is not the case since $\ln(x - 10)$ is defined for $x > 10$.

6. Critical points: To find critical points, we need to find where the derivative of $F(x)$ is equal to zero. $F'(x) = 2x \ln(x - 10) + \frac{x^2}{x - 10}$ Setting $F'(x) = 0$ and solving for $x$: $2x \ln(x - 10) + \frac{x^2}{x - 10} = 0$ Since the natural logarithm of a negative number is undefined, this equation has no real solutions for $x$ in the domain $x > 10$.

7. Increasing/Decreasing Intervals: To determine the intervals of increasing and decreasing, we need to examine the sign of the derivative $F'(x)$. However, as there are no critical points in the domain $x > 10$, we can conclude that the function is either strictly increasing or strictly decreasing throughout its domain. To determine which case, we would need additional information about the behavior of the function.

8. Concavity: The concavity of the function $F(x)$ can be determined by analyzing the sign of its second derivative $F''(x)$. $F''(x) = 2 \ln(x - 10) + \frac{4x}{(x - 10)^2}$ To find inflection points, we need to find where $F''(x) = 0$. However, similarly to the critical points, this equation has no real solutions for $x$ in the domain $x > 10$.

Graphically, the function $F(x) = x^2 \ln(x - 10)$ will exist only for $x > 10$ and has a y-intercept at the origin (0, 0). However, without additional information, it is challenging to determine the exact behavior of the function and its concavity throughout its domain.