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Solution for Find the Interval in Which the Following Function Are Increasing Or Decreasing F(X) = X3 − 6x2 + 9x + 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) = x3 − 6x2 + 9x + 15 ?

Solution

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

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

$f\left( x \right) = x^3 - 6 x^2 + 9x + 15$

$f'\left( x \right) = 3 x^2 - 12x + 9$

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

$= 3 \left( x - 1 \right)\left( x - 3 \right)$

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

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

$\Rightarrow 3 \left( x - 1 \right)\left( x - 3 \right) > 0$

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

$\Rightarrow x < 1 \ or \ x > 3$

$\Rightarrow x \in \left( - \infty , 1 \right) \cup \left( 3, \infty \right)$

$\text { So,f(x)is increasing on } x \in \left( - \infty , 1 \right) \cup \left( 3, \infty \right).$

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

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

$\Rightarrow 3 \left( x - 1 \right)\left( x - 3 \right) < 0$

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

$\Rightarrow 1 < x < 3$

$\Rightarrow x \in \left( 1, 3 \right)$

$\text { So,f(x)is decreasing on x } \in \left( 1, 3 \right) .$

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Solution for question: Find the Interval in Which the Following Function Are Increasing Or Decreasing F(X) = X3 − 6x2 + 9x + 15 ? concept: Increasing and Decreasing Functions. For the courses CBSE (Science), PUC Karnataka Science, CBSE (Arts), CBSE (Commerce)
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