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 $f'(x) = 0$

Vedclass · Accepted Answer

$f: R \rightarrow R$ 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.

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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