The real valued function $f(x) = x^{15} + 5x^9 + 10$ is increasing for___________.

A function is increasing when $f'(x) \geq 0$.

Given: $f(x) = x^{15} + 5x^9 + 10$

Using the power rule:

$f'(x) = 15x^{14} + 45x^8$

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Factoring out common terms:

$f'(x) = 15x^8(x^6 + 3)$

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Analyzing the sign of $f'(x) = 15x^8(x^6 + 3)$:

The factor $15$ is always positive.

The factor $x^8$ is always non-negative for all real $x$ (even power).

The factor $(x^6 + 3)$ is always positive since $x^6 \geq 0$, which means $x^6 + 3 \geq 3 > 0$.

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Therefore:

$f'(x) = (\text{positive}) \times (\text{non-negative}) \times (\text{positive}) \geq 0$

This holds for all real values of $x$.

Note that $f'(x) = 0$ only when $x = 0$, and $f'(x) > 0$ for all $x \neq 0$.

Since $f'(x) \geq 0$ for all real $x$, the function is increasing for all real values of $x$.