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# Solution for Find the Interval in Which the Following Function Are Increasing Or Decreasing F(X) = X3 − 12x2 + 36x + 17 ? - CBSE (Commerce) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

#### Question

Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?

#### Solution

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

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

$f\left( x \right) = x^3 - 12 x^2 + 36x + 17$

$f'\left( x \right) = 3 x^2 - 24x + 36$

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

$= 3 \left( x - 2 \right)\left( x - 6 \right)$

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

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

$\Rightarrow 3 \left( x - 2 \right)\left( x - 6 \right) > 0$

$\Rightarrow \left( x - 2 \right)\left( x - 6 \right) > 0 \left[ \text { Since } 3 > 0, 3 \left( x - 2 \right)\left( x - 6 \right) > 0 \Rightarrow \left( x - 2 \right)\left( x - 6 \right) > 0 \right]$

$\Rightarrow x < 2 orx > 6$

$\Rightarrow x \in \left( - \infty , 2 \right) \cup \left( 6, \infty \right)$

$\text { So },f(x)\text { is increasing on } x \in \left( - \infty , 2 \right) \cup \left( 6, \infty \right).$

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

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

$\Rightarrow 3 \left( x - 2 \right)\left( x - 6 \right) < 0$

$\Rightarrow \left( x - 2 \right)\left( x - 6 \right) < 0 \left[ \text { Since } 3 > 0, 3 \left( x - 2 \right)\left( x - 6 \right) < 0 \Rightarrow \left( x - 2 \right)\left( x - 6 \right) < 0 \right]$

$\Rightarrow 2 < x < 6$

$\Rightarrow x \in \left( 2, 6 \right)$

$\text { So,}f(x)\text { is decreasing on } x \in \left( 2, 6 \right) .$

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Solution Find the Interval in Which the Following Function Are Increasing Or Decreasing F(X) = X3 − 12x2 + 36x + 17 ? Concept: Increasing and Decreasing Functions.
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