14 multiple choice 1 point graph of f the figure above shows the graph of a function f. which of the…

Answer:

E. $\frac{f(b)-f(a)}{b - a}$

Explanation:

Step1: Analyze $f(a)$

$f(a)$ is a function - value at $x = a$, which is negative as the graph is below the $x$-axis at $x=a$.

Step2: Analyze $f^{\prime}(a)$

The derivative $f^{\prime}(a)$ represents the slope of the tangent line at $x = a$. The slope of the tangent at $x=a$ is positive since the function is increasing near $x = a$.

Step3: Analyze $f^{\prime}(c)$

The derivative $f^{\prime}(c)$ represents the slope of the tangent line at $x = c$. At the maximum - point of the function, the slope of the tangent is $f^{\prime}(c)=0$.

Step4: Analyze $f(d)-f(c)$

$f(d)-f(c)$ is the change in the function - value from $x = c$ to $x = d$. Since the function is decreasing from $x = c$ to $x = d$, $f(d)-f(c)<0$.

Step5: Analyze $\frac{f(b)-f(a)}{b - a}$

$\frac{f(b)-f(a)}{b - a}$ is the average rate of change of the function from $x = a$ to $x = b$. Geometrically, it is the slope of the secant line connecting the points $(a,f(a))$ and $(b,f(b))$. The secant line from $(a,f(a))$ to $(b,f(b))$ has a positive slope, and by comparing with the other values: $f(a)<0$, $f^{\prime}(c) = 0$, $f(d)-f(c)<0$, and $f^{\prime}(a)$ is positive but the secant - line slope $\frac{f(b)-f(a)}{b - a}$ is steeper (has a larger positive value) than the slope of the tangent at $x = a$. So $\frac{f(b)-f(a)}{b - a}$ has the greatest value.