rompt: Consider the equation $x^2 + \frac{9}{x^2} = 10$. Solve for $x$ such that the inequality $x^3 - 3x > 0$ is also satisfied. Additionally, show that $y = x^3 - 3x$ is increasing at $x = 1$.

To solve the given equation 
x
2
+
9
x
2
=
10
x 
2
 + 
x 
2
 
9
​
 =10, let's start by multiplying both sides of the equation by 
x
2
x 
2
  to get rid of the fraction. This gives us 
x
4
+
9
=
10
x
2
x 
4
 +9=10x 
2
 .



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Rearranging the equation, we get 
x
4
−
10
x
2
+
9
=
0
x 
4
 −10x 
2
 +9=0. This is a quadratic equation in terms of 
x
2
x 
2
 , so we can use the quadratic formula to solve for 
x
2
x 
2
 .



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

The quadratic formula is given by 
x
=
−
b
±
b
2
−
4
a
c
2
a
x= 
2a
−b± 
b 
2
 −4ac
​
 
​
 . In this case, 
a
=
1
a=1, 
b
=
−
10
b=−10, and 
c
=
9
c=9. Plugging these values into the formula, we get 
x
2
=
10
±
(
−
10
)
2
−
4
(
1
)
(
9
)
2
(
1
)
x 
2
 = 
2(1)
10± 
(−10) 
2
 −4(1)(9)
​
 
​
 .



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Simplifying the expression, we get 
x
2
=
10
±
100
−
36
2
=
10
±
64
2
=
10
±
8
2
x 
2
 = 
2
10± 
100−36
​
 
​
 = 
2
10± 
64
​
 
​
 = 
2
10±8
​
 .



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

This gives us two possible values for 
x
2
x 
2
 : 
x
2
=
10
+
8
2
=
9
x 
2
 = 
2
10+8
​
 =9 and 
x
2
=
10
−
8
2
=
1
x 
2
 = 
2
10−8
​
 =1.



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Taking the square root of both sides, we get 
x
=
±
3
x=±3 and 
x
=
±
1
x=±1.



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Now, we need to check which of these solutions satisfy the inequality 
x
3
−
3
x
>
0
x 
3
 −3x>0.



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

We can start by plugging in 
x
=
3
x=3 into the inequality: 
3
3
−
3
(
3
)
=
27
−
9
=
18
>
0
3 
3
 −3(3)=27−9=18>0. So, 
x
=
3
x=3 satisfies the inequality.



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Next, we plug in 
x
=
−
3
x=−3: 
(
−
3
)
3
−
3
(
−
3
)
=
−
27
+
9
=
−
18
>
0
(−3) 
3
 −3(−3)=−27+9=−18>0. So, 
x
=
−
3
x=−3 satisfies the inequality.



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Now, we plug in 
x
=
1
x=1: 
1
3
−
3
(
1
)
=
1
−
3
=
−
2
<
0
1 
3
 −3(1)=1−3=−2<0. So, 
x
=
1
x=1 does not satisfy the inequality.



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Finally, we plug in 
x
=
−
1
x=−1: 
(
−
1
)
3
−
3
(
−
1
)
=
−
1
+
3
=
2
>
0
(−1) 
3
 −3(−1)=−1+3=2>0. So, 
x
=
−
1
x=−1 satisfies the inequality.



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Therefore, the solutions to the equation that also satisfy the inequality are 
x
=
3
x=3 and 
x
=
−
1
x=−1.



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Now, let's show that 
y
=
x
3
−
3
x
y=x 
3
 −3x is increasing at 
x
=
1
x=1. To do this, we need to find the derivative of 
y
y with respect to 
x
x and evaluate it at 
x
=
1
x=1.



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Using the power rule of differentiation, we get 
d
y
d
x
=
3
x
2
−
3
dx
dy
​
 =3x 
2
 −3.



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Evaluating the derivative at 
x
=
1
x=1, we get 
d
y
d
x
(
1
)
=
3
(
1
)
2
−
3
=
3
−
3
=
0
dx
dy
​
 (1)=3(1) 
2
 −3=3−3=0.



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

However, to show that the function is increasing, we need to look at the behavior of the derivative around 
x
=
1
x=1. We can do this by looking at the derivative at 
x
=
1
+
ϵ
x=1+ϵ for some small 
ϵ
>
0
ϵ>0.



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate