Question

F(x) = x^2ln(x^2-10)​

Answer 1

Ответ:

Объяснение:

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

Answer 2

To analyze the function $F(x) = x^2 \ln(x^2 - 10)$, we'll look at its properties step by step.

1. Domain: The function is defined for real numbers $x$ such that $x^2 - 10 > 0$ since the natural logarithm is only defined for positive arguments. So, we need to solve the inequality $x^2 - 10 > 0$: $x^2 - 10 > 0$ $x^2 > 10$ $x > \sqrt{10}$ (Note that $x$ can also be less than the negative value of $\sqrt{10}$, but since the square of $x$ appears in the function $F(x)$, we only consider positive values of $x$ to avoid duplication.)

Thus, the domain of $F(x)$ is $x \in (\sqrt{10}, +\infty)$.

2. Critical points: To find critical points, we need to find the values of $x$ where the derivative of $F(x)$ is equal to zero or undefined.

The derivative of $F(x)$ with respect to $x$ can be found using the product and chain rules of differentiation: $F'(x) = \frac{d}{dx} \left( x^2 \ln(x^2 - 10) \right) = x^2 \cdot \frac{1}{x^2 - 10} + 2x \ln(x^2 - 10)$

Now, we set $F'(x) = 0$ and solve for critical points: $x^2 \cdot \frac{1}{x^2 - 10} + 2x \ln(x^2 - 10) = 0$

There isn't a straightforward way to analytically solve for $x$ in this equation. We can use numerical methods or graphical analysis to find approximate critical points.

3. Behavior near the asymptote: The natural logarithm has an asymptote at $x = 0$. When $x$ approaches $0$ from the right, $\ln(x^2 - 10)$ approaches negative infinity, and thus, $F(x)$ approaches negative infinity. As $x$ approaches $\sqrt{10}$ from the right (the lower bound of the domain), $\ln(x^2 - 10)$ approaches negative infinity as well, and $F(x)$ approaches negative infinity.

4. Behavior as $x \rightarrow \infty$: As $x$ approaches infinity, the behavior of $F(x)$ will be dominated by the term $x^2 \ln(x^2)$, as $-10$ becomes negligible compared to large values of $x$. The natural logarithm grows slowly compared to any power of $x$ (e.g., $x^2$), so the function $F(x)$ will tend to infinity as $x$ approaches infinity.

To summarize:

• Domain: $x \in (\sqrt{10}, +\infty)$
• Critical points: Need to use numerical or graphical methods to find approximate values.
• Asymptotes: Vertical asymptote at $x = 0$ due to the natural logarithm. Approaches negative infinity as $x$ approaches $\sqrt{10}$ from the right.
• Behavior as $x \rightarrow \infty$: $F(x)$ tends to infinity.

Please note that without knowing the exact critical points, we can't fully analyze the behavior of the function in detail. Numerical or graphical methods can be used to find the critical points and get a more comprehensive understanding of the function.