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Question
Let f(x) = x3 − 6x2 + 15x + 3. Then,
Options
f(x) > 0 for all x ∈ R
f(x) > f(x + 1) for all x ∈ R
f(x) is invertible
none of these
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Solution
f(x) is invertible
f(x) =x3 − 6x2 + 15x + 3
\[f'(x) = 3 x^2 - 12x + 15\]
\[ = 3\left( x^2 - 4x + 5 \right)\]
\[ = 3\left( x^2 - 4x + 4 + 1 \right)\]
\[ = 3 \left( x - 2 \right)^2 + \frac{1}{3} > 0\]
\[\text { Therefore, f(x) is strictly increasing function }. \]
\[ \Rightarrow f^{- 1} (x) \text { exists } . \]
\[\text { Hence, f(x) is an invertible function } .\]
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