Share

Books Shortlist
Your shortlist is empty

# Solution for Find the Interval in Which the Following Function Are Increasing Or Decreasing F(X) = X4 − 4x3 + 4x2 + 15 ? - CBSE (Science) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

#### Question

Find the interval in which the following function are increasing or decreasing  f(x) = x4 − 4x3 + 4x2 + 15 ?

#### Solution

$\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.$

$\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .$

$f\left( x \right) = x^4 - 4 x^3 + 4 x^2 + 15$

$f'\left( x \right) = 4 x^3 - 12 x^2 + 8x$

$= 4x \left( x^2 - 3x + 2 \right)$

$= 4x \left( x - 1 \right)\left( x - 2 \right)$

$\text { Here, 0, 1 and 2 are the critical points }.$

$\text { The possible intervals are }\left( - \infty , 0 \right),\left( 0, 1 \right),\left( 1, 2 \right)\text { and }\left( 2, \infty \right). ...(1)$

$\text { For f(x) to be increasing, we must have}$

$f'\left( x \right) > 0$

$\Rightarrow 4x \left( x - 1 \right)\left( x - 2 \right) > 0 \left[ \text { Since } 4 > 0, 4x \left( x - 1 \right)\left( x - 2 \right) > 0 \Rightarrow x \left( x - 1 \right)\left( x - 2 \right) > 0 \right]$

$\Rightarrow x \left( x - 1 \right)\left( x - 2 \right) > 0$

$\Rightarrow x \in \left( 0, 1 \right) \cup \left( 2, \infty \right) \left[ \text { From eq }. (1) \right]$

$\text { So },f(x)\text { is increasing on x } \in \left( 0, 1 \right) \cup \left( 2, \infty \right) .$

$\text { For }f(x) \text { to be decreasing, we must have }$

$f'\left( x \right) < 0$

$\Rightarrow 4x \left( x - 1 \right)\left( x - 2 \right) < 0 \left[ \text { Since } 4 > 0, 4x \left( x - 1 \right)\left( x - 2 \right) < 0 \Rightarrow x \left( x - 1 \right)\left( x - 2 \right) < 0 \right]$

$\Rightarrow x \left( x - 1 \right)\left( x - 2 \right) < 0$

$\Rightarrow x \in \left( - \infty , 0 \right) \cup \left( 1, 2 \right) \left[ \text { From eq. } (1) \right]$

$\text { So,}f(x)\text { is decreasing on x } \in \left( - \infty , 0 \right) \cup \left( 1, 2 \right) .$

Is there an error in this question or solution?

#### Video TutorialsVIEW ALL [3]

Solution Find the Interval in Which the Following Function Are Increasing Or Decreasing F(X) = X4 − 4x3 + 4x2 + 15 ? Concept: Increasing and Decreasing Functions.
S