Let h(x) = frac{5(f(x))^3}{3} + frac{(f(x))^2 | vedclass

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(C) Given $h(x) = \frac{5}{3}(f(x))^3 + \frac{1}{2}(f(x))^2 + 2f(x) + 100$. To determine the behavior of $h(x)$,we differentiate with respect to $x$:

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$h(x)$ increases as $f(x)$ increases

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(C) Given $h(x) = \frac{5}{3}(f(x))^3 + \frac{1}{2}(f(x))^2 + 2f(x) + 100$. To determine the behavior of $h(x)$,we differentiate with respect to $x$: $h'(x) = \frac{d}{dx} \left[ \frac{5}{3}(f(x))^3 + \frac{1}{2}(f(x))^2 + 2f(x) + 100 \right]$. Using the chain rule: $h'(x) = \left( 5(f(x))^2 + f(x) + 2 \right) f'(x)$. Consider the quadratic expression $Q = 5(f(x))^2 + f(x) + 2$. The discriminant $D = b^2 - 4ac = (1)^2 - 4(5)(2) = 1 - 40 = -39$. Since $D  0)$,the quadratic $Q$ is always positive for all real values of $f(x)$. Therefore,the sign of $h'(x)$ depends solely on the sign of $f'(x)$. If $f'(x) > 0$,then $h'(x) > 0$,meaning $h(x)$ increases as $f(x)$ increases. Thus,$h(x)$ increases as $f(x)$ increases.

Let $h(x) = \frac{5(f(x))^3}{3} + \frac{(f(x))^2}{2} + 2f(x) + 100$,where $f(x)$ is a differentiable function. Then which one of the following is correct?

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