Question

If f(x) = $x^3-10x^2+200x-10$, then $f\left(x\right)$ is-

• A. decreasing in $(-\infty ,10]$ and increasing in $(10,\infty )$
• B. increasing in $(-\infty ,10]$ decreasing in $(10,\infty )$
• C. increasing for every value of $x$
• D. decreasing for every value of $x$

Answer 1

The correct option is C increasing for every value of $x$

$f^{\prime }\left(x\right)=3x^2-20x+200>0$
Now consider
$g\left(x\right)=3x^2-20x+200=0$
$D=400-600$
$=-200$
Hence
$D<0$
$f^{\prime }\left(x\right)>0$ for all $x.$
Hence
$f\left(x\right)>0$ for all $xϵR$.