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Solution for Find the Interval in Which the Following Function Are Increasing Or Decreasing F(X) = 5 + 36x + 3x2 − 2x3 ? - CBSE (Science) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

Question

Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x?

Solution

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

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

$f\left( x \right) = 5 + 36x + 3 x^2 - 2 x^3$

$f'\left( x \right) = 36 + 6x - 6 x^2$

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

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

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

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

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

$\Rightarrow \left( x - 3 \right)\left( x + 2 \right) < 0 \left[ \text {Since} - 6 < 0, - 6 \left( x - 1 \right)\left( x + 2 \right) > 0 \Rightarrow \left( x - 1 \right)\left( x + 2 \right) < 0 \right]$

$\Rightarrow - 2 < x < 3$

$\Rightarrow x \in \left( - 2, 3 \right)$

$\text { So },f(x)\text { is increasing on } \left( - 2, 3 \right) .$

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

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

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

$\Rightarrow \left( x - 3 \right)\left( x + 2 \right) > 0 \left[ \text { Since } - 6 < 0, - 6 \left( x - 1 \right)\left( x + 2 \right) < 0 \Rightarrow \left( x - 1 \right)\left( x + 2 \right) > 0 \right]$

$\Rightarrow x < - 2 \ or \ x > 3$

$\Rightarrow x \in \left( - \infty , - 2 \right) \cup \left( 3, \infty \right)$

$\text { So,}f(x)\text { is decreasing on } \left( - \infty , - 2 \right) \cup \left( 3, \infty \right) .$

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Solution for question: Find the Interval in Which the Following Function Are Increasing Or Decreasing F(X) = 5 + 36x + 3x2 − 2x3 ? concept: Increasing and Decreasing Functions. For the courses CBSE (Science), CBSE (Commerce), CBSE (Arts), PUC Karnataka Science
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