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

Question

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:

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