Question

draw the graph of the function ( y = f(x) ) that satisfies the following properties. the function decreases and the slope increases as ( x ) increases. select all graphs that satisfy both properties. a. b. c. d.

Explanation:

• For a function \(y = f(x)\) that decreases, the slope \(f^{\prime}(x)<0\).
• For the slope to increase as \(x\) increases, the second - derivative \(f^{\prime\prime}(x)>0\) (since \(f^{\prime\prime}(x)=\frac{d}{dx}(f^{\prime}(x))\) and we want \(f^{\prime}(x)\) to become less negative or increase).
• Graph A: The function is decreasing (\(y\) - values are getting smaller as \(x\) increases) and the slope is increasing (the curve is concave up, which means \(f^{\prime\prime}(x)>0\)).
• Graph B: The function is increasing (\(y\) - values are getting larger as \(x\) increases), so it does not satisfy the "function decreases" property.
• Graph C: The function is increasing (\(y\) - values are getting larger as \(x\) increases) for most of the domain (after a certain point), so it does not satisfy the "function decreases" property.
• Graph D: The function is decreasing (\(y\) - values are getting smaller as \(x\) increases) but the slope is decreasing (the curve is concave down, \(f^{\prime\prime}(x)<0\)).

Answer:

A.