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Find the Interval in Which the Following Function Are Increasing Or Decreasing F ( X ) = X 4 4 + 2 3 X 3 − 5 2 X 2 − 6 X + 7 - Mathematics

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Question

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?

Sum
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Solution

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

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

\[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\]

\[ = \frac{3 x^4 + 8 x^3 - 30 x^2 - 72x + 84}{12}\]

\[f'\left( x \right) = \frac{12 x^3 + 24 x^2 - 60x - 72}{12}\]

\[ = \left( x^3 + 2 x^2 - 5x - 6 \right)\]

\[ = \left( x + 1 \right)\left( x^2 + x - 6 \right)\]

\[ = \left( x + 1 \right)\left( x - 2 \right)\left( x + 3 \right)\]

\[\text { Here }, -1, 2 \text { and } -3 \text { are the critical points }.\]

\[\text { The possible intervals are }\left( - \infty - 3 \right),\left( - 3, - 1 \right),\left( - 1, 2 \right)and\left( 2, \infty \right).\]

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

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

\[ \Rightarrow \left( x + 1 \right)\left( x - 2 \right)\left( x + 3 \right) > 0\]

\[ \Rightarrow x \in \left( - 3, - 1 \right) \cup \left( 2, \infty \right)\]

\[\text { So },f(x)\text { is increasing on } x \in \left( - 3, - 1 \right) \cup \left( 2, \infty \right) . \]

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

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

\[ \Rightarrow \left( x + 1 \right)\left( x - 2 \right)\left( x + 3 \right) < 0\]

\[ \Rightarrow x \in \left( - \infty - 3 \right) \cup \left( - 1, 2 \right) \left[ \text { From eq }. (1) \right]\]

\[\text { So,}f(x)\text { is decreasing on x } \in \left( - \infty - 3 \right) \cup \left( - 1, 2 \right) .\]

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Chapter 17: Increasing and Decreasing Functions - Exercise 17.2 [Page 33]

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RD Sharma Mathematics [English] Class 12
Chapter 17 Increasing and Decreasing Functions
Exercise 17.2 | Q 1.2 | Page 33

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