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Find the Interval in Which the Following Function Are Increasing Or Decreasing F(X) = X2 + 2x − 5 ?

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प्रश्न

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

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उत्तर

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

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

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

\[f'\left( x \right) = 2x + 2\]

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

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

\[ \Rightarrow 2x + 2 > 0\]

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

\[ \Rightarrow x + 1 > 0\]

\[ \Rightarrow x > - 1\]

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

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

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

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

\[ \Rightarrow 2x + 2 < 0\]

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

\[ \Rightarrow x + 1 < 0\]

\[ \Rightarrow x < - 1\]

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

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

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अध्याय 16: Increasing and Decreasing Functions - Exercise 17.2 [पृष्ठ ३३]

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आर.डी. शर्मा Mathematics Volume 1 and 2 [English] Class 12
अध्याय 16 Increasing and Decreasing Functions
Exercise 17.2 | Q 1.02 | पृष्ठ ३३

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