1 - (0 , 0) and (2 , 8). (i). Explain without calculations...

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CALCULUS 1501 WINTER 2010 HOMEWORK ASSIGNMENT 1. Due January 15. 1.1. Formulate and prove a statement similar to Lemma 1.8 for the case when f 0 ( x 0 ) < 0. (See Lecture 1 of the Course Notes ). 1.2. Give an example of a function which is continuous on the interval ( -∞ , 0] but does not attain a maximum or a minimum value. 1.3. Prove that if a nonconstant function f ( x ) satisfies the conditions of Rolle’s theorem on the interval [ a,b ], then there exist points x 1 and x 2 on the interval ( a,b ) such that f 0 ( x 1 ) < 0 and f 0 ( x 2 ) > 0. 1.4. On the interval (0 , 2) there exists a point c such that the tangent line to the graph of the function y = x 3 at the point ( c,c 3 ) is parallel to the straight line passing through the points
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Unformatted text preview: (0 , 0) and (2 , 8). (i). Explain without calculations why such point c necessarily exists. (ii). Find c . 1.5. Prove using the Mean Value Theorem: x 1 + x &lt; ln(1 + x ) , for x &gt; 0. 1.6. Prove using the Mean Value Theorem: e x &gt; 1 + x + x 2 2 , for x &gt; 0. 1.7. Show that the equation x 4 + 4 x + c = 0 has at most two real roots. Here c is an arbitrary constant. (Hint: argue by contradiction - suppose that there are three dierent roots. Now try to use Rolles theorem.) [This is Problem 20 in Section 4.2 of the textbook (p. 286).] 1...
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