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

all real values of x

The correct answer is Option (4) → all real values of x

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

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

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

$x^{8}\ge 0$ for all real $x$ and $(x^{6}+3)>0$ for all real $x$.

$\Rightarrow f'(x)\ge 0$ for every real $x$, and equals $0$ only at $x=0$.

Therefore $f(x)$ is strictly increasing on all real numbers.

The function is increasing for all real values of $x$.