The problem in the image asks to prove that the derivative of the function:

$f(x) = (x - 3)(x - 2)(x + 2)(x + 3)$

has only real roots.

Step-by-Step Solution:

Step 1: Simplifying the Function

Let's simplify the function $f(x)$ first:

$f(x) = (x - 3)(x - 2)(x + 2)(x + 3)$

We can group and multiply the factors:

[ f(x) = [(x - 3)(x + 3)] \cdot [(x - 2)(x + 2)] ]

Using the identity $(a - b)(a + b) = a^2 - b^2$, we get:

$f(x) = (x^2 - 9)(x^2 - 4)$

Now, expand this product:

$f(x) = x^4 - 4x^2 - 9x^2 + 36 = x^4 - 13x^2 + 36$

So the simplified form of the function is:

$f(x) = x^4 - 13x^2 + 36$

Step 2: Deriving the Function

Now, let's take the derivative of $f(x)$:

$f'(x) = \frac{d}{dx}(x^4 - 13x^2 + 36)$

Using basic derivative rules:

$f'(x) = 4x^3 - 26x$

Thus, the derivative is:

$f'(x) = 4x(x^2 - 6.5)$

Step 3: Finding the Roots of the Derivative

To find the roots of the derivative $f'(x)$, we solve:

$4x(x^2 - 6.5) = 0$

This gives two cases:

1. $4x = 0$ ⟹ $x = 0$
2. $x^2 - 6.5 = 0$ ⟹ $x^2 = 6.5$ ⟹ $x = \pm \sqrt{6.5}$

The approximate roots are $x = \pm 2.55$.

Step 4: Conclusion

The roots of the derivative are $x = 0$, $x = 2.55$, and $x = -2.55$, which are all real numbers. Thus, the derivative of the function has only real roots.

Would you like more details or have any questions?

Here are 5 related questions for further understanding:

1. How does the identity $(a - b)(a + b) = a^2 - b^2$ simplify polynomial expressions?
2. What are the rules for differentiating polynomial functions?
3. How can we apply Rolle's theorem to verify the existence of roots between intervals?
4. How does the sign of the second derivative help in determining concavity?
5. How can we check if a function has complex roots without directly solving for them?

Tip: When solving for roots, factor as much as possible to simplify the expression before using advanced methods.