To determine which of the given functions have both increasing and decreasing intervals, we analyze their derivatives.
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\( f(x) = \sqrt{4x} \):
- This is a square root function, and its domain is \( x \geq 0 \).
- Its derivative \( f'(x) = \frac{2}{\sqrt{4x}} = \frac{1}{\sqrt{x}} \).
- The derivative is always positive for \( x > 0 \), which means this function is always increasing in its domain.
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\( f(x) = -x^2 + 5 \):
- This is a quadratic function that opens downward.
- Its derivative \( f'(x) = -2x \).
- Setting the derivative to 0 gives \( x = 0 \).
- The sign of the derivative changes from positive (increasing) for \( x < 0 \) to negative (decreasing) for \( x > 0 \).
- Therefore, this function has both increasing and decreasing intervals.
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\( f(x) = 7.5^x \):
- This is an exponential function.
- Its derivative \( f'(x) = 7.5^x \ln(7.5) \).
- Since \( 7.5^x > 0 \) and \( \ln(7.5) > 0 \), the derivative is always positive, indicating that this function is always increasing.
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\( f(x) = -x - 7 \):
- This is a linear function.
- Its derivative \( f'(x) = -1 \), which is always negative.
- This function is always decreasing.
Based on this analysis, the function that has both increasing and decreasing intervals is:
\( f(x) = -x^2 + 5 \).