Question

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

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 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.