Which of the following statements could be tr | vedclass

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(D) Given $f''(x) = x^{1/3}$.Integrating with respect to $x$:$f'(x) = \int x^{1/3} dx = \frac{x^{4/3}}{4/3} + C_1 = \frac{3}{4}x^{4/3} + C_1$.Comparin

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$I$ & $IV$ only

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(D) Given $f''(x) = x^{1/3}$.Integrating with respect to $x$:$f'(x) = \int x^{1/3} dx = \frac{x^{4/3}}{4/3} + C_1 = \frac{3}{4}x^{4/3} + C_1$.Comparing this with the given statements:Statement $I$ $(f'(x) = \frac{3}{4}x^{4/3} + 9)$ is possible if $C_1 = 9$.Statement $IV$ $(f'(x) = \frac{3}{4}x^{4/3} - 4)$ is possible if $C_1 = -4$.Now,integrating $f'(x)$ to find $f(x)$:$f(x) = \int (\frac{3}{4}x^{4/3} + C_1) dx = \frac{3}{4} \cdot \frac{x^{7/3}}{7/3} + C_1x + C_2 = \frac{9}{28}x^{7/3} + C_1x + C_2$.Statements $II$ and $III$ involve $f(x) = \frac{9}{28}x^{7/3} + K$. This form is only possible if $C_1 = 0$. If $C_1 = 0$,then $f'(x) = \frac{3}{4}x^{4/3}$,which does not match $I$ or $IV$. Thus,$II$ and $III$ cannot be true simultaneously with the provided $f'(x)$ options. Therefore,only $I$ and $IV$ are valid forms for $f'(x)$.

$I$. $f'(x) = \frac{3}{4}x^{4/3} + 9$	$II$. $f(x) = \frac{9}{28}x^{7/3} - 2$
$III$. $f(x) = \frac{9}{28}x^{7/3} + 6$	$IV$. $f'(x) = \frac{3}{4}x^{4/3} - 4$

Which of the following statements could be true if $f''(x) = x^{1/3}$?

$I$. $f'(x) = \frac{3}{4}x^{4/3} + 9$ $II$. $f(x) = \frac{9}{28}x^{7/3} - 2$

$III$. $f(x) = \frac{9}{28}x^{7/3} + 6$ $IV$. $f'(x) = \frac{3}{4}x^{4/3} - 4$

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Which of the following statements could be true if $f''(x) = x^{1/3}$? $I$. $f'(x) = \frac{3}{4}x^{4/3} + 9$ $II$. $f(x) = \frac{9}{28}x^{7/3} - 2$ $III$. $f(x) = \frac{9}{28}x^{7/3} + 6$ $IV$. $f'(x) = \frac{3}{4}x^{4/3} - 4$