Rolle's Theorem Solved Numericals Explained in Hindi l Engineering Mathematics

Summary

In this five-minute engineering video, the host dives into numerical problems, aiming to enhance viewers' understanding of solving polynomial functions. The video focuses on the function f(x) = x² - 5x + 4, explaining the importance of continuity and differentiability within the closed interval [1, 4]. It demonstrates how to find the function's value at specific points and its derivatives, ultimately applying the Rolle's Theorem to deduce the existence of a critical point between the given interval. The host encourages viewers to like, share, and watch the video for a deeper dive into numerical concepts.

Takeaways

• 🔧 The video introduces a tutorial on solving numerical problems related to engineering concepts.
• 📝 The function f(x) = x² - 5x + 4 is provided, and it is continuous over the closed interval [1, 4].
• ✅ The function is also differentiable over the open interval (1, 4), as the derivative f'(x) = 2x - 5 exists and is continuous.
• 🔍 The video checks the condition f(a) = f(b), where a = 1 and b = 4, confirming that f(1) = 0 and f(4) = 0, thus they are equal.
• 🎯 According to Rolle's Theorem, there exists a value c in (1, 4) such that f'(c) = 0. The video calculates c as 2.5, which lies within the interval.
• 🔄 The video repeats the process for another function, f(x) = x², over the interval [-1, 1].
• 💡 It is confirmed that f(x) = x² is continuous and differentiable over the interval, with f(-1) = 1 and f(1) = 1, satisfying the conditions of Rolle's Theorem.
• 🧮 The value of c is found to be 0 in this case, which also lies within the interval [-1, 1].
• 📜 The video provides a clear explanation of how to apply Rolle's Theorem to specific polynomial functions.
• 👍 The video concludes by encouraging viewers to like and share the content if they found it helpful.

Q & A

What is the main topic of the video?

The main topic of the video is to demonstrate and explain the process of solving a quadratic equation and understanding the properties of a polynomial function within a closed interval.

What is the given polynomial function in the video?

The given polynomial function is f(x) = x^2 - 5x + 4.

What are the values of 'a' and 'b' in the context of the video?

In the context of the video, 'a' is given as 1 and 'b' is given as 4, referring to the closed interval [1, 4].

What is the first condition to check for the function in the video?

The first condition to check is whether the function is continuous, which means it should exist and have a value at every point in the closed interval [1, 4].

Is the function in the video differentiable within the open interval (a, b)?

Yes, the function is differentiable within the open interval (1, 4) as its derivative is 2x - 5, which exists and is continuous in the given interval.

How to find the value of 'a' or 'b' where the function equals zero?

To find the value where the function equals zero, set the function equal to zero and solve for 'x'. In this case, x^2 - 5x + 4 = 0 can be solved to find the roots.

What is the derivative of the given polynomial function?

The derivative of the polynomial function f(x) = x^2 - 5x + 4 is 2x - 5.

What does the video suggest to do after finding the derivative of the function?

After finding the derivative, the video suggests to set it equal to zero to find the critical points, which are potential points where the function could have a local maximum or minimum.

How does the video explain the relationship between the derivative and the function's extremum?

The video explains that if the derivative equals zero, it indicates a potential extremum. By solving 2x - 5 = 0, we find the critical point 'c' which is 2.5, which lies between 'a' and 'b'.

What is the conclusion of the video regarding the function's extremum?

The conclusion is that the function has an extremum at 'c' = 2.5, which is within the interval [1, 4], and this extremum is a minimum since the derivative changes sign around this point.

What is the final message of the video to the viewers?

The final message is to encourage viewers to like the video if they found it helpful and to share it with friends, thanking them for watching.