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Question: In this problem we prove a special case of the Mean Value Theorem where f(a) = f(b) = 0. This special case is called Rolle's Theorem: If f is continuous on [a, b] and differentiable on (a, b), and if f(a) = f(b) = 0, then there is a number c, with a
f'(c) = 0
By the Extreme Value Theorem, f has a global maximum and a global minimum on [a, b].
(a) Prove Rolle's Theorem in the case that both the global maximum and the global minimum are at endpoints of [a, b]. [Hint: f(x) must be a very simple function in this case.]
(b) Prove Rolle's Theorem in the case that either the global maximum or the global minimum is not at an endpoint.
A book company spends $1,000 per week on rent for its factory. Each book made at the factory requires $2 worth of paper and $8 worth of labour and energy. Answer the following questions assuming that the factory produces 500 books per week.
The ratio between the length and breadth of a rectangular field is 11:7. The cost of fencing it is Rs. 75,000. Find the dimensions of the field.
Find the principal components of the data for Exercise 2.
Find the average of x2 + y2 over the triangle with vertices (0, 0), (1, 0), and (1, 1). Find the coordinates of the centroid of the region bounded by the curves y = 2x2 and y=12 - x2.
Write the null and alternative hypotheses
things vary in life - virtually nothing except physical standards such as the speed of light we interact with is
Vail Ski Shop received a $1,209 invoice dated July 16 with 2/10, 1/15, n/60 terms. On July 30, Vail sent a $493 partial payment.
John steps on an escalator to leave a subway. Then, 5 seconds later, Jane steps on a parallel escalator to enter the subway. They pass 14 seconds after that. If both escalators are travelling at a speed of 5 feet per second, what is the length of ..
Let a composite number n = pq has two prime factors. Prove that n cannot be a Carmichael number. Alice wants to send the plaintext "CRYFTOLOGY" to Bob via RSA cryptosystem, using Bob's public key. Bob then has to decrypt it. How can she implement i..
What is the principle of powers? Describe in your own words. When solving a radical equation, how do we use the principle of powers to eliminate radicals? Demonstrate with an example.
1. After reviewing their SWOT analysis, assuming the role of a manager, what would be one decision you would recommend based on the strategic plan and the SWOT analysis?
problem 1evaluate where c is the arc of the parabola y2 4x with initial point 1-2 and terminal point 12.problem
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