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Question
Find the interval in which the following function are increasing or decreasing f(x) = x2 + 2x − 5 ?
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Solution
\[\text { When }\left( x - a \right)\left( x - b \right)>0 \text { with} a < b, x < a \ or \ x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = x^2 + 2x - 5\]
\[f'\left( x \right) = 2x + 2\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 2x + 2 > 0\]
\[ \Rightarrow 2\left( x + 1 \right) > 0\]
\[ \Rightarrow x + 1 > 0\]
\[ \Rightarrow x > - 1\]
\[ \Rightarrow x \in \left( - 1, \infty \right)\]
\[\text { So,}f(x)\text { is increasing on } \left( - 1, \infty \right) . \]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 2x + 2 < 0\]
\[ \Rightarrow 2\left( x + 1 \right) < 0\]
\[ \Rightarrow x + 1 < 0\]
\[ \Rightarrow x < - 1\]
\[ \Rightarrow x \in \left( - \infty , - 1 \right)\]
\[\text { So,}f(x)\text { is decreasing on }\left( - \infty , - 1 \right).\]
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