Question

Which of the following functions will have both an increasing and a decreasing interval?(1 point)
Responses

f(x)=4x−−√
f left parenthesis x right parenthesis equals 4 Start Root x End Root

f(x)=−x2+5
f left parenthesis x right parenthesis equals negative x squared plus 5

f(x)=7.5x
f left parenthesis x right parenthesis equals 7.5 superscript x baseline

f(x)=−x−7

Answer 1

To determine which of the given functions have both increasing and decreasing intervals, we analyze their derivatives.

1. **\( f(x) = \sqrt{4x} \)**:
- 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.

2. **\( f(x) = -x^2 + 5 \)**:
- 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.

3. **\( f(x) = 7.5^x \)**:
- 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.

4. **\( f(x) = -x - 7 \)**:
- 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:

**\( f(x) = -x^2 + 5 \)**.