Answer:
• Mean Slope = -7
,
• c=0.5
Explanation:
Given the function:
[tex]f\mleft(x\mright)=6-7x^2[/tex]
Part A
We want to find the mean slope on the interval [-6, 7].
First, evaluate f(7) and f(-6):
[tex]\begin{gathered} f(7)=6-7(7^2)=6-7(49)=6-343=-337 \\ f(-6)=6-7(-6)^2=6-7(36)=6-252=-246 \end{gathered}[/tex]
Next, substitute these values into the formula for the mean slope.
[tex]\begin{gathered} \text{ Mean Slope}=\frac{f(7)-f(-6)}{7-(-6)}=\frac{-337-(-246)}{7+6}=\frac{-337+246}{13} \\ =-\frac{91}{13} \\ =-7 \end{gathered}[/tex]
The mean slope of the function over the interval [-6,7] is -7.
Part B
Given the function, f(x):
[tex]f\mleft(x\mright)=6-7x^2[/tex]
Its derivative, f'(x) will be:
[tex]f^{\prime}(x)=-14x[/tex]
Replace c for x:
[tex]f^{\prime}(c)=-14c[/tex]
Equate f'(c) to the mean slope obtained in part a.
[tex]-14c=-7[/tex]
Solve for c:
[tex]\begin{gathered} \frac{-14c}{-14}=\frac{-7}{-14} \\ c=0.5 \end{gathered}[/tex]
The value of c that satisfies the mean value theorem is 0.5.
