If f(x) = x^3 - 10 x^2 + 200x - 10, then - Vidyadip

MCQ

If $f(x) = {x^3} - 10{x^2} + 200x - 10$, then

A
$f(x)$ is decreasing in $[ - \infty ,10]$ and increasing in $[10,\,\infty ]$
B
$f(x)$ is increasing in $[ - \infty ,10]$ and decreasing in $[10,\,\infty ]$
✓
$f(x)$ is increasing throughout real line
D
$f(x)$ is decreasing throughout real line

✓

Answer

Correct option: C.

$f(x)$ is increasing throughout real line

c
(c) $f(x) = {x^3} - 10{x^2} + 200x - 10$

$f'(x) = 3{x^2} - 20x + 200$

For increasing $f'(x) > 0$ ==> $3{x^2} - 20x + 200 > 0$

$3{\rm{ }}\left[ {{x^2} - \frac{{20}}{3}x + \frac{{200}}{3} + \frac{{100}}{9} - \frac{{100}}{9}} \right] > 0$

$ \Rightarrow 3{\rm{ }}\left[ {{{\left( {x - \frac{{10}}{3}} \right)}^2} + \frac{{500}}{9}} \right] $

$0$ $ \Rightarrow 3{\rm{ }}{\left( {x - \frac{{10}}{3}} \right)^2} + \frac{{500}}{3} > 0$

Always increasing throughout real line.

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