The real valued function $f(x) = x^{15} +5x^9 + 10$ is increasing for ______.
Answer & explanation
Correct answer: option 4
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 $x$.