On which intervals is the function $f (x) = x^{3} - 3 x^{2} + 4 x - 5$ increasing?

easy 1 mark Class 12 · Mathematics · Application of Derivatives

Answer

(- in f t y, in f t y)

The correct answer is $(- in f t y, in f t y)$ — Conclude that $f (x)$ is increasing on the entire real line, i.e. $(- in f t y, in f t y)$

Step-by-step solution

Differentiate $f (x)$ : $f^{'} (x) = 3 x^{2} - 6 x + 4$
Set $f^{'} (x) > 0$ for increasing intervals
Solve $3 x^{2} - 6 x + 4 > 0$ by examining the quadratic: discriminant $D e l t a = (- 6)^{2} - 4 c d o t 3 c d o t 4 = 36 - 48 = - 12 < 0$
Since $D e l t a < 0$ and the leading coefficient $3 > 0$ , $f^{'} (x)$ is positive for all real $x$
Conclude that $f (x)$ is increasing on the entire real line, i.e. $(- in f t y, in f t y)$

Common mistake: Ignoring the sign of the derivative or miscalculating the discriminant of a quadratic

Concept tested: Determine intervals of increase using derivative