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Solution for Find the Interval in Which the Following Function Are Increasing Or Decreasing F(X) = 2x3 − 15x2 + 36x + 1 ? - CBSE (Commerce) Class 12 - Mathematics

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Question

Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?

Solution

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

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

\[ f\left( x \right) = 2 x^3 - 15 x^2 + 36x + 1\]

\[f'\left( x \right) = 6 x^2 - 30x + 36\]

\[ = 6 \left( x^2 - 5x + 6 \right)\]

\[ = 6 \left( x - 2 \right)\left( x - 3 \right)\]

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

\[f'\left( x \right) > 0\]

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

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

\[ \Rightarrow x < 2 \ or \ x > 3\]

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

\[\text { So },f(x)\text { is increasing on } x \in \left( - \infty , 2 \right) \cup \left( 3, \infty \right).\]

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

\[f'\left( x \right) < 0\]

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

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

\[ \Rightarrow 2 < x < 3 \]

\[ \Rightarrow x \in \left( 2, 3 \right)\]

\[\text { So },f(x)\text { is decreasing on } x \in \left( 2, 3 \right) .\]

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Solution for question: Find the Interval in Which the Following Function Are Increasing Or Decreasing F(X) = 2x3 − 15x2 + 36x + 1 ? concept: Increasing and Decreasing Functions. For the courses CBSE (Commerce), CBSE (Arts), PUC Karnataka Science, CBSE (Science)
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