To analyze the functions provided, \( f(x) = 2\sqrt{x} \) and \( g(x) = 2 \cdot \sqrt[3]{x} \), we can examine their derivatives to determine whether they are increasing or decreasing.
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Function \( f(x) = 2\sqrt{x} \):
- The derivative can be found using the power rule: \[ f'(x) = 2 \cdot \frac{1}{2} x^{-1/2} = \frac{1}{\sqrt{x}}. \]
- The derivative \( f'(x) \) is positive for all \( x > 0 \) (since the square root function is defined for non-negative \( x \), we consider \( x \geq 0 \)). Therefore, \( f(x) \) is an increasing function for \( x \geq 0 \).
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Function \( g(x) = 2\cdot \sqrt[3]{x} \):
- The derivative is also found using the power rule: \[ g'(x) = 2 \cdot \frac{1}{3} x^{-2/3} = \frac{2}{3\sqrt[3]{x^2}}. \]
- The derivative \( g'(x) \) is positive for all \( x \) (including negative values because \( \sqrt[3]{x} \) is defined for all real numbers). Therefore, \( g(x) \) is also an increasing function.
In conclusion, both functions \( f(x) \) and \( g(x) \) are increasing.
Thus, the correct response is:
Both graphs are increasing.