Q. 2 Fill in the blanks to complete e... [FREE SOLUTION]

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Chapter 3: Q. 2 (page 313)

Fill in the blanks to complete each of the following theorem statements:
Rolle's Theorem: If $f$ is _ on $[a, b]$ and on localid="1648528803641" $(a, b)$ , and if , then there exists at least one value $c 鈭� (a, b)$ for which $f' (c) =$ __.

Short Answer

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Ans:

Rolle's Theorem: If $f$ is continuous on $[a, b]$ and differentiableonrole="math" localid="1648528826355" $(a, b)$ , and ifrole="math" localid="1648528898168" $f (a) = f (b) = 0$ , then there exists at least one value $c 鈭� (a, b)$ for which role="math" localid="1648528834973" $f' (x) = 0 .$

Step by step solution

Step 1. Given information,

Rolle鈥檚 Theorem

Step 2. Explanation:

Rolle鈥檚 Theorem is an immediate consequence of the Extreme Value Theorem and the fact that every extremum is a critical point. Suppose f is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$ , with $f (a) = f (b) = 0$ . By the Extreme Value Theorem, we know that f attains both a maximum and a minimum value on $[a, b]$ . If one of these extreme values occurs at a point $x = c$ in the interior $(a, b)$ of the interval, the $x = c$ is a local extremum of $f$ . By the previous theorem, this means that $x = c$ is a critical point of $f$ . Since f is assumed to be differentiable at $x = c$ , it follows that $f' (c) = 0$ and we are done.

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91影视

Fill in the blanks to complete each of the following theorem statements:
Rolle's Theorem: If $f$ is _ on $[a, b]$ and on localid="1648528803641" $(a, b)$ , and if , then there exists at least one value $c 鈭� (a, b)$ for which $f' (c) =$ __.

Short Answer

Step by step solution

Step 1. Given information,

Step 2. Explanation:

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91影视

Fill in the blanks to complete each of the following theorem statements:Rolle's Theorem: If fis _____ on [a,b]and ______ on localid="1648528803641" (a,b), and if ______ , then there exists at least one value c鈭�(a,b)for which f'(c)=______.

Short Answer

Step by step solution

Step 1. Given information,

Step 2. Explanation:

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Discrete Mathematics

Geometry

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.

Fill in the blanks to complete each of the following theorem statements:
Rolle's Theorem: If $f$ is _ on $[a, b]$ and on localid="1648528803641" $(a, b)$ , and if , then there exists at least one value $c 鈭� (a, b)$ for which $f' (c) =$ __.