For the cubic function f(x) = 2x^3 + 9x^2 + 1 | vedclass

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(C) Given $f(x) = 2x^3 + 9x^2 + 12x + 1$.First,find the derivative: $f'(x) = 6x^2 + 18x + 12 = 6(x^2 + 3x + 2) = 6(x + 2)(x + 1)$.Setting $f'(x) = 0$

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$f: R \rightarrow R$ is bijective

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(C) Given $f(x) = 2x^3 + 9x^2 + 12x + 1$.First,find the derivative: $f'(x) = 6x^2 + 18x + 12 = 6(x^2 + 3x + 2) = 6(x + 2)(x + 1)$.Setting $f'(x) = 0$ gives critical points $x = -2$ and $x = -1$.For $x  0$ (increasing).For $-2 For $x > -1$,$f'(x) > 0$ (increasing).Thus,the function is non-monotonic (it increases,then decreases,then increases).Next,find the second derivative for the inflection point: $f''(x) = 12x + 18$. Setting $f''(x) = 0$ gives $x = -18/12 = -3/2$.Since the function has local maxima and minima,it is a many-to-one function and therefore not bijective.Statement $A$ is true (it is non-monotonic).Statement $B$ is true (based on the intervals of increase/decrease).Statement $D$ is true (inflection point at $x = -3/2$).Statement $C$ is false because the function is not bijective.

List-$I$	List-$II$
$A$. If $y = \|x\| + \|x - 2\|$,then at $x = 2$,$\frac{dy}{dx} =$	$I$. $2$
$B$. If $f(x) = \|\cos 2x\|$,then $f'(\frac{\pi}{4} +) =$	$II$. $0$
$C$. If $f(x) = \sin(\pi[x])$,where $[x]$ is the greatest integer function,then $f'(1-) =$	$III$. $-2$
$D$. If $f(x) = \log\|x - 1\|$,$x \neq 1$,then $f'(\frac{1}{2}) =$	$IV$. does not exist

For the cubic function $f(x) = 2x^3 + 9x^2 + 12x + 1$,which one of the following statements does not hold true?

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