QuestionGiven $h(x) = g(x+1) - g(x) = -6$ , which statement about $f$ and $g$ must be true? (A, B, C, or D)

Studdy Solution

STEP 1

Assumptions1. The function $f$ is not explicitly given. . The function $g$ is defined as $g(x)=f(x+1)-f(x)$ .
3. The function $h$ is defined as $h(x)=g(x+1)-g(x)$ .
4. $h(x)=-6$ for all values of $x$ .

STEP 2

First, we need to express $h(x)$ in terms of $f(x)$ . We know that $h(x)=g(x+1)-g(x)$ , and we know that $g(x)=f(x+1)-f(x)$ . So we can substitute $g(x)$ into the equation for $h(x)$ .
$h(x) = g(x+1) - g(x) = [f((x+1)+1) - f(x+1)] - [f(x+1) - f(x)]$

STEP 3

implify the equation for $h(x)$ .
$h(x) = f(x+2) -2f(x+1) + f(x)$

STEP 4

Given that $h(x)=-6$ for all values of $x$ , we can set the equation for $h(x)$ equal to $-6$ .
$f(x+2) -2f(x+1) + f(x) = -6$

STEP 5

This equation tells us that the second difference of the function $f$ is constant and equal to $-$ . In other words, the function $f$ is a quadratic function with a second coefficient of $-3$ (since the second coefficient is half of the second difference).

STEP 6

Since the second coefficient of a quadratic function determines the direction of its concavity, we can conclude that the graph of $f$ is concave down.

STEP 7

The function $g$ is the first difference of $f$ , which corresponds to the slope of the tangent line to the graph of $f$ . Because $f$ is concave down, its slope decreases as $x$ increases. Therefore, $g$ is a decreasing function.

STEP 8

From the above analysis, we can conclude that option () is correct "Because $h$ is negative and constant, $g$ is decreasing, and the graph of $f$ is concave down."

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