which of the following statements is true about the exponential function ( h ) given by ( h(x)=-3 cdot 4^{x}…

which of the following statements is true about the exponential function ( h ) given by ( h(x)=-3 cdot 4^{x} )?\n\na ( h ) is always increasing, and the graph of ( h ) is always concave up.\n\nb ( h ) is always increasing, and the graph of ( h ) is always concave down.\n\nc ( h ) is always decreasing, and the graph of ( h ) is always concave up.\n\nd ( h ) is always decreasing, and the graph of ( h ) is always concave down.

which of the following statements is true about the exponential function ( h ) given by ( h(x)=-3 cdot 4^{x} )?\n\na ( h ) is always increasing, and the graph of ( h ) is always concave up.\n\nb ( h ) is always increasing, and the graph of ( h ) is always concave down.\n\nc ( h ) is always decreasing, and the graph of ( h ) is always concave up.\n\nd ( h ) is always decreasing, and the graph of ( h ) is always concave down.

Answer

Explanation:

Step1: Analyze the first - derivative

The function is (h(x)=-3\cdot4^{x}). The derivative of (y = a\cdot b^{x}) is (y^\prime=a\cdot b^{x}\ln b). For (h(x)), (h^\prime(x)=-3\cdot4^{x}\ln4). Since (4^{x}>0) and (\ln4>0), then (h^\prime(x)<0) for all (x). So (h(x)) is always decreasing.

Step2: Analyze the second - derivative

The second - derivative (h^{\prime\prime}(x)=-3\cdot4^{x}(\ln4)^{2}). Because (4^{x}>0) and ((\ln4)^{2}>0), then (h^{\prime\prime}(x)<0) for all (x). A function with (h^{\prime\prime}(x)<0) for all (x) has a graph that is always concave down.

Answer:

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