Question

1. (1 point) the graph of g is shown below. sketch the graph of g.

Explanation:

Step1: Identify the function type

The graph of \(g(x)\) is a straight - line. The general form of a straight - line function is \(y = mx + b\). Using two points \((0,2)\) and \((1,0)\), the slope \(m=\frac{0 - 2}{1-0}=- 2\) and the \(y\) - intercept \(b = 2\), so \(g(x)=-2x + 2\).

Step2: Find the first - derivative

The derivative of \(g(x)=-2x + 2\) using the power rule \((x^n)^\prime=nx^{n - 1}\) is \(g^\prime(x)=\frac{d}{dx}(-2x + 2)=-2\).

Step3: Find the second - derivative

The derivative of a constant function \(y = c\) (where \(c=-2\) in the case of \(g^\prime(x)\)) is \(0\). So \(g^{\prime\prime}(x)=\frac{d}{dx}(-2)=0\).

Answer:

The graph of \(g^{\prime\prime}(x)\) is a horizontal line at \(y = 0\) for all \(x\) values.