Question

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

• A. $f\left(x\right)$ decreasing in $[-\infty ,10]$ and increasing in $[10,\infty ]$
• B. $f\left(x\right)$ increasing in $[-\alpha ,10]$ and decreasing in $[10,\alpha ]$
• C. $f\left(x\right)$ in increasing throughout real line
• D. None of these

Answer 1

Answer: (C)

$f\left(x\right)$ in increasing throughout real line

$f(x0=x^3-10x^2+200x-10\Rightarrow f^{\prime }(x)=3x^2-20x+200$
But $3x^2-20x+200=3\left[x^2-\frac{20}{3}x+\frac{200}{3}\right]$
$=3[{\left(x-\frac{10}{3}\right)}^2+\frac{200}{3}-\frac{100}{9}]$
$=[{(x-\frac{10}{3})}^2+\frac{500}{9}]$ > 0 for all x
Therefore, f is increasing throughout real line