Question

the figure show graphs of $f$, $f$, and $f$. identify each curve. (description: figure with three graphs. graph \a\ as it progresses right rises through the 3rd quadrant into the 2nd quadrant, turns downward and falls intersecting the negative x - axis, continuing downward intersecting the negative y - axis, continues to fall and turns upward in the 4th quadrant and rises through the 1st quadrant. graph \b\ falls through the 2nd quadrant, continuing to fall as it progress right intersecting the negative x - axis, turns upward as it intersects the negative y - axis, and continues to rise through the 1st quadrant. graph \c\ is a rising line. the graph rises from the 3rd quadrant, intersects the 2nd quadrant, then the positive x - axis and continue to rise through the 1st quadrant.) $a = f$, $b = f$, $c = f$, $a = f$, $b = f$, $c = f$, $a = f$, $b = f$, $c = f$, $a = f$, $b = f$, $c = f$

Explanation:

Step1: Recall derivative - function relationship

The derivative $f'$ represents the slope of the function $f$. The second - derivative $f''$ represents the concavity of $f$. If $f$ is increasing, $f'>0$; if $f$ is decreasing, $f'<0$. If $f$ is concave up, $f''>0$; if $f$ is concave down, $f''<0$.

Step2: Analyze the linear graph

The graph of $c$ is a straight - line with a positive slope. A linear function has a constant non - zero first - derivative and a zero second - derivative. The slope of a linear function is constant, so it cannot be the original function $f$ (since $f$ is not linear) and it cannot be $f''$ (because $f''$ for a non - linear function is not constant in most cases). So $c = f'$.

Step3: Analyze the concavity

The graph of $b$ is concave up everywhere (it has a U - shape). For a function that is concave up, $f''>0$. Since $c = f'$ and $c$ is positive (the line $c$ has a positive slope), and $b$ is concave up, $b=f$.

Step4: Determine $f''$

Since $b = f$ and $c = f'$, then $a=f''$. We can also note that when $a>0$, $b$ is concave up, which is consistent with the relationship between the second - derivative and concavity.

Answer:

$a = f'',b = f,c = f'$