Question

the function f is not explicitly given. the function g is given by g(x)=f(x + 1)-f(x). the function h is given by h(x)=g(x + 1)-g(x). if h(x)=-6 for all values of x, which of the following statements must be true? a because h is negative and constant, the graphs of g and f always have negative slope. b because h is negative and constant, the graphs of g and f are concave down. c because h is negative and constant, g is decreasing, and the graph of f always has negative slope. d because h is negative and constant, g is decreasing, and the graph of f is concave down.

Explanation:

Step1: Recall the relationship between first - and second - differences and derivatives

The function \(g(x)=f(x + 1)-f(x)\) can be thought of as a discrete - difference approximation of the derivative of \(f\). And \(h(x)=g(x + 1)-g(x)\) is a discrete - difference approximation of the second - derivative of \(f\). In the continuous case, if \(y = f(x)\), the first derivative \(f^\prime(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}\), and for \(h = 1\), \(g(x)\approx f^\prime(x)\) (a forward - difference approximation). Also, \(h(x)\) is related to the second - derivative \(f^{\prime\prime}(x)\). Since \(h(x)=g(x + 1)-g(x)\), \(h(x)\) is like the rate of change of \(g(x)\).

Step2: Analyze the sign of \(h(x)\)

We are given that \(h(x)=- 6\) for all \(x\). Since \(h(x)=g(x + 1)-g(x)<0\) for all \(x\), this means that \(g(x + 1)<g(x)\) for all \(x\). So the function \(g(x)\) is a decreasing function.

Step3: Relate \(h(x)\) to the concavity of \(f(x)\)

In the continuous case, the second - derivative of a function \(y = f(x)\) determines the concavity of the function. A negative second - derivative \(f^{\prime\prime}(x)<0\) implies that the function \(y = f(x)\) is concave down. Since \(h(x)\) is a discrete analogue of the second - derivative of \(f(x)\) and \(h(x)=-6<0\) for all \(x\), the graph of \(f(x)\) is concave down.

Answer:

D. Because \(h\) is negative and constant, \(g\) is decreasing, and the graph of \(f\) is concave down.