Question

1 which of the following statements is true about the exponential function h given by h(x) = -3·4ˣ? a h is always increasing, and the graph of h is always concave up. b h is always increasing, and the graph of h is always concave down. c h is always decreasing, and the graph of h is always concave up. d h is always decreasing, and the graph of h is always concave down.

Explanation:

Step1: Analyze the exponential function

The function is \( h(x) = -3 \cdot 4^x \). For an exponential function \( y = a \cdot b^x \), if \( a>0 \) and \( b > 1 \), it's increasing; if \( a<0 \) and \( b>1 \), it's decreasing. Here, \( a=-3<0 \) and \( b = 4>1 \), so \( h(x) \) is always decreasing.

Step2: Analyze concavity

The second derivative (or the nature of exponential functions) shows that for \( y = a \cdot b^x \), the first derivative is \( y'=a \cdot b^x \ln b \), the second derivative is \( y''=a \cdot b^x (\ln b)^2 \). Since \( a=-3 \), \( b = 4 \), \( (\ln b)^2>0 \), so \( y''=-3 \cdot 4^x (\ln 4)^2<0 \), which means the function is concave down.

Answer:

D. \( h \) is always decreasing, and the graph of \( h \) is always concave down.