Questions: Consider the function f(x)=7-5x^2 on the interval [-4,6]. Find the average rate of change of the function on this interval, i.e. (f(6)-f(-4))/(6-(-4))= By the Mean Value Theorem, we know there exists a c in the open interval (-4,6) such that f'(c) is equal to this average rate of change. For this problem, there is only one c that works. Find this value of c. c=

Consider the function f(x)=7-5x^2 on the interval [-4,6]. Find the average rate of change of the function on this interval, i.e.
(f(6)-f(-4))/(6-(-4))=

By the Mean Value Theorem, we know there exists a c in the open interval (-4,6) such that f'(c) is equal to this average rate of change. For this problem, there is only one c that works. Find this value of c.
c=

Transcript text: 1. Submit answer Get help Practice similar Consider the function $f(x)=7-5 x^{2}$ on the interval $[-4,6]$. Find the average rate of change of the function on this interval, i.e. \[ \frac{f(6)-f(-4)}{6-(-4)}= \] $\square$ By the Mean Value Theorem, we know there exists a $c$ in the open interval $(-4,6)$ such that $f^{\prime}(c)$ is equal to this average rate of change. For this problem, there is only one $c$ that works. Find this value of $c$. \[ c= \] $\square$ Submit answer Next item

Solution

Solution Steps

Step 1: Calculate the Average Rate of Change

The average rate of change of the function on the interval $[-4, 6]$ is calculated as: \[\frac{f(6)-f(-4)}{6+4} = -10\]

Step 2: Find the Derivative of the Function

The derivative of the function $f(x)$ is: \[f^\prime(x) = - 10 x\]

Step 3: Solve for $c$

Solve the equation $f^\prime(x) = -10$ for $x$ to find the value(s) of $c$ in the open interval $(-4, 6)$. The value(s) of $c$ that satisfy the condition are: $c = 1$.

Final Answer:

The value(s) of $c$ in the open interval $(-4, 6)$ that satisfy the Mean Value Theorem are: $c = 1$.

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