The problem you uploaded asks for two things:

1. Solve the equation: $x^2 - \frac{9}{x^2} = 10$ for $x$ such that the inequality $x^3 - 3x > 0$ is also satisfied.

2. Show that the function: $y = x^3 - 3x$ is increasing at $x = 1$.

Step 1: Solving the equation

The equation is: $x^2 - \frac{9}{x^2} = 10$ Multiplying both sides by $x^2$ to eliminate the fraction gives: $x^4 - 9 = 10x^2$ Rearranging this, we get a quadratic equation in terms of $x^2$: $x^4 - 10x^2 - 9 = 0$ Let $z = x^2$. This simplifies to: $z^2 - 10z - 9 = 0$ We can solve this quadratic using the quadratic formula: $z = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(-9)}}{2(1)}$ $z = \frac{10 \pm \sqrt{100 + 36}}{2} = \frac{10 \pm \sqrt{136}}{2}$ $z = \frac{10 \pm 2\sqrt{34}}{2}$ $z = 5 \pm \sqrt{34}$ Thus, $x^2 = 5 + \sqrt{34}$ or $x^2 = 5 - \sqrt{34}$.

Since $\sqrt{34}$ is approximately 5.83, the second solution $5 - \sqrt{34}$ is negative, which is not possible for $x^2$. Hence, the only valid solution is: $x^2 = 5 + \sqrt{34}$ Taking the square root of both sides: $x = \pm \sqrt{5 + \sqrt{34}}$

Step 2: Checking the inequality $x^3 - 3x > 0$

The inequality to check is $x^3 - 3x > 0$. Factorizing this, we get: $x(x^2 - 3) > 0$ The critical points occur when $x = 0$ or $x = \pm \sqrt{3}$.

Testing intervals:

• For $x \in (-\infty, -\sqrt{3})$, the expression is positive.
• For $x \in (-\sqrt{3}, 0)$, the expression is negative.
• For $x \in (0, \sqrt{3})$, the expression is negative.
• For $x \in (\sqrt{3}, \infty)$, the expression is positive.

Thus, $x^3 - 3x > 0$ when $x > \sqrt{3}$ or $x < -\sqrt{3}$.

Step 3: Verifying at $x = 1$ that the function is increasing

To show that $y = x^3 - 3x$ is increasing at $x = 1$, we take the derivative: $y' = \frac{d}{dx}(x^3 - 3x) = 3x^2 - 3$ At $x = 1$: $y'(1) = 3(1)^2 - 3 = 3 - 3 = 0$ This means the function is not increasing or decreasing at $x = 1$, as the derivative equals zero (it has a critical point).

Would you like more details or have any further questions?

Here are 5 related questions to explore further:

1. How do we solve quadratic equations in general?
2. What does the sign of the derivative tell us about a function's behavior?
3. What is the graphical interpretation of the function $y = x^3 - 3x$?
4. How can we determine intervals where a function is increasing or decreasing?
5. How do we solve equations involving higher powers (quartic equations)?

Tip: Always verify the domain of solutions when dealing with square roots or fractions in equations!