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: Analyze \( h(x) \) and \( g(x) \)

Given \( h(x)=g(x + 1)-g(x)=-6 \) for all \( x \). This means \( g(x + 1)=g(x)-6 \), so as \( x \) increases by 1, \( g(x) \) decreases by 6. Thus, \( g \) is a decreasing function (since the difference \( g(x + 1)-g(x) \) is negative, indicating \( g \) decreases as \( x \) increases).

Step2: Analyze concavity of \( f(x) \)

Recall that \( g(x)=f(x + 1)-f(x) \). Then \( h(x)=g(x + 1)-g(x)=[f(x + 2)-f(x + 1)]-[f(x + 1)-f(x)]=f(x + 2)-2f(x + 1)+f(x) \). For a function \( y = f(x) \), the second - difference (in the context of discrete functions, analogous to the second derivative in continuous functions) \( f(x + 2)-2f(x + 1)+f(x) \) being negative and constant implies that the function \( f(x) \) is concave down. A negative second - difference (or a negative second derivative in the continuous case) indicates concave down behavior.

Now let's analyze each option:

• Option A: Just because \( h(x) \) is negative and constant, we can't say \( f \) always has a negative slope. The slope of \( f \) is related to \( g(x) \), but \( g(x) \) could be positive or negative. For example, if \( g(x)=10 - 6x \), \( g(x) \) is decreasing, but for small \( x \), \( g(x) \) is positive (so \( f(x + 1)-f(x)>0 \), meaning \( f \) has a positive slope at those points). So A is wrong.
• Option B: We know \( g(x) \) is decreasing (from \( h(x)=g(x + 1)-g(x)=-6<0 \)), but the statement says "the graphs of \( g \) and \( f \) are concave down". For \( g(x) \), since \( h(x)=g(x + 1)-g(x)=-6 \) (constant), the second - difference of \( g(x) \) is 0 (because \( h(x + 1)-h(x)=-6-(-6) = 0 \)), so \( g(x) \) is a linear function (not concave down, it's linear). So B is wrong.
• Option C: As we saw in Option A, \( f \) does not always have a negative slope. So C is wrong.
• Option D: We showed that \( g(x) \) is decreasing (since \( g(x + 1)-g(x)=-6<0 \)) and \( f(x) \) is concave down (since \( h(x)=f(x + 2)-2f(x + 1)+f(x)=-6<0 \) for all \( x \)). So D is correct.

Answer:

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