Solving Functions Problems: Lagrange and Rolle's Theorem

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In summary, the author of the question presents two possible statements, a and b, and argues that b is the correct answer based on Lagrange's theorem. However, the author overlooks the fact that the hypotheses of Rolle's theorem are satisfied, allowing for the possibility of statement a being true. Upon further examination, it can be concluded that both statements are in fact true.
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laura1231
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Hi, in a book I have found this problem:
"Let be $f,g:\mathbb{R}\rightarrow\mathbb{R}$ two derivable functions such that $f(0)=g(0)$ and $f(6)=g(6)$. Which of the following statements is necessarily true?:
a) $\exists\ c\in]0;6[ : f'(c)=g'(c)$;
b) $\exists\ c_1,c_2\in]0;6[ : f'(c_1)=g'(c_2)$.
"
The author of this question indicates the answer b because, for Lagrange's theorem $\exists\ c_1\in ]0;6[ : f'(c_1)=\dfrac{f(6)-f(0)}{6-0}$ and $\exists\ c_2\in ]0;6[ : g'(c_2)=\dfrac{g(6)-g(0)}{6-0}$, therefore $f'(c_1)=g'(c_2)$ but you can't be sure that $c_1=c_2$...
I think the author misses in fact if you call $h(x)=f(x)-g(x)$ then, the hypothesis of Rolle's theorem are true ($h(0)=h(6)$) therefore $\exists\ c \in ]0;6[: h'(c)=0$ then $f'(c)=g'(c)$. For me the correct answer is a.
Am I right?
 
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laura123 said:
Hi, in a book I have found this problem:
"Let be $f,g:\mathbb{R}\rightarrow\mathbb{R}$ two derivable functions such that $f(0)=g(0)$ and $f(6)=g(6)$. Which of the following statements is necessarily true?:
a) $\exists\ c\in]0;6[ : f'(c)=g'(c)$;
b) $\exists\ c_1,c_2\in]0;6[ : f'(c_1)=g'(c_2)$.
"
The author of this question indicates the answer b because, for Lagrange's theorem $\exists\ c_1\in ]0;6[ : f'(c_1)=\dfrac{f(6)-f(0)}{6-0}$ and $\exists\ c_2\in ]0;6[ : g'(c_2)=\dfrac{g(6)-g(0)}{6-0}$, therefore $f'(c_1)=g'(c_2)$ but you can't be sure that $c_1=c_2$...
I think the author misses in fact if you call $h(x)=f(x)-g(x)$ then, the hypothesis of Rolle's theorem are true ($h(0)=h(6)$) therefore $\exists\ c \in ]0;6[: h'(c)=0$ then $f'(c)=g'(c)$. For me the correct answer is a.
Am I right?

Hi laura! ;)

Yep. I believe you are right.
Therefore both statements are true.
 

FAQ: Solving Functions Problems: Lagrange and Rolle's Theorem

What is Lagrange's Theorem and how is it used to solve function problems?

Lagrange's Theorem, also known as the Mean Value Theorem, states that for a function that is continuous on a closed interval and differentiable on the open interval, there will be at least one point in the open interval where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints of the interval. This point is known as a critical point and can be used to find the maximum and minimum values of the function.

What is Rolle's Theorem and how is it related to Lagrange's Theorem?

Rolle's Theorem is a special case of Lagrange's Theorem where the function has a critical point in the open interval and the slopes of the tangent and secant lines are equal at this point. In other words, Rolle's Theorem states that if a function is continuous on a closed interval, differentiable on the open interval, and has the same values at the endpoints of the interval, there will be at least one point in the open interval where the derivative of the function is equal to zero.

How do you use Lagrange's Theorem to find the maximum and minimum values of a function?

To use Lagrange's Theorem, you first need to find the critical points of the function by setting the derivative of the function equal to zero and solving for x. Then, plug these critical points into the original function to find the corresponding y-values. The largest y-value will be the maximum value of the function, and the smallest y-value will be the minimum value.

Can Lagrange's Theorem be used to solve problems involving multivariable functions?

Yes, Lagrange's Theorem can be extended to solve problems involving multivariable functions. In this case, the critical points are found by setting the partial derivatives of the function with respect to each variable equal to zero. These critical points can then be used to find the maximum and minimum values of the function.

What are some real-life applications of Lagrange's Theorem and Rolle's Theorem?

Lagrange's Theorem and Rolle's Theorem have various applications in fields such as physics, economics, and engineering. For example, they can be used to determine the maximum and minimum values of a function representing the trajectory of a projectile or the profit function of a company. They can also be used to optimize the design of structures by finding the maximum and minimum values of a function representing the stress or strain on the structure in different locations.

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