Can someone help me better understand how to solve this problem?

[Zgrahamy]

I know that flat out begging someone for an answer is not the way to master Calculus. I just need help with the overall methodology behind solving problems like this one.

 

[Dr.Peterson]

View attachment 37684 I know that flat out begging someone for an answer is not the way to master Calculus. I just need help with the overall methodology behind solving problems like this one.

This is an illustration of the Mean Value Theorem.

They first tell you something you could have figured out yourself; the average rate of change on [0, 4] is
[math]\frac{f(4)-f(0)}{4-0}=\frac{((4)^2-7(4)+3)-((0)^2-7(0)+3)}{4-0}=\frac{(-9)-(3)}{4}=-3[/math]
The theorem says that there will be some value [imath]x=c[/imath] within the interval [0, 4] where the derivative (instantaneous rate of change) equals -3. They want you to find that value.

So, find the derivative of [imath]f(x)[/imath], set it equal to -3, and solve.

 

[Steven G]

...and when you solve for f(x) = -3 only use results that lie within [0,4]

 

[khansaheb]

you solve for f(x) = -3

I assume, you meant f'(x) = -3