Question

the figure shows graphs of f, f, and f. identify each curve. (description: three graphs on one grid. graph \a\ is line with negative slope through the origin. graph \b\ is downward parabola with vertex on the positive x - axis. graph \c\ has two turns. it falls from from the 2nd quadrant down into the 3rd quadrant; turns upward and rises intersecting the negative y - axis into the 4th quadrant continuing into the 1st quadrant. the graph then turns downward and falls into and through the 4th quadrant.)
o a = f, b = f, c = f
o a = f, b = f, c = f
o a = f, b = f, c = f
o a = f, b = f, c = f
o no correct answer choice is given.

Explanation:

Step1: Analyze the properties of the first - derivative

The first - derivative $f'$ gives the slope of the function $f$. A linear function has a constant slope. The parabola (graph b) has a linear first - derivative because the slope of a parabola changes linearly. The function $f$ (graph c) has a non - constant slope, and its first - derivative should be a non - constant function. Since graph a is a straight line with a negative slope, it cannot be the first - derivative of a non - linear function like graph c. So graph b is likely $f'$ as the slope of graph c changes in a way that a parabola can represent.

Step2: Analyze the properties of the second - derivative

The second - derivative $f''$ gives the rate of change of the first - derivative. The first - derivative (graph b, a parabola) has a non - constant rate of change. The rate of change of a parabola is a linear function. Graph a is a straight line, which can represent the rate of change of a parabola (graph b). So graph a is $f''$. And graph c is the original function $f$.

Answer:

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