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प्रश्न
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
विकल्प
x < −3
| x | > 3
x ≤ −3
| x | ≥ 3
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उत्तर
Function f(x) = x3 − 27x + 5 is monotonically increasing when | x | > 3.
Explanation:
\[f\left( x \right) = x^3 - 27x + 5\]
\[f'\left( x \right) = 3 x^2 - 27\]
\[ = 3 \left( x^2 - 9 \right)\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 3 \left( x^2 - 9 \right) > 0\]
\[ \Rightarrow \left( x^2 - 9 \right) > 0 \left[ \text { Since } 3 > 0, 3 \left( x^2 - 9 \right) > 0 \Rightarrow \left( x^2 - 9 \right) > 0| \right]\]
\[ \Rightarrow \left( x + 3 \right)\left( x - 3 \right) > 0\]
\[ \Rightarrow x < - 3 \ or \ x > 3\]
\[ \Rightarrow \left| x \right| > 3\]
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