How do you check the hypotheses of Rolles Theorem and the Mean Value Theorem and find a value of c that makes the appropriate conclusion true for #f(x) = x^3+x^2#?
The hypotheses and conclusions for both of those theorems involve a function on an interval.
Without knowing the interval we cannot test the third hypothesis for Rolle's Theorem, nor can we state the conclusions for the theorems.
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To check the hypotheses of Rolle's Theorem and the Mean Value Theorem for ( f(x) = x^3 + x^2 ):

For Rolle's Theorem: a. Verify that ( f(x) ) is continuous on the closed interval ([a, b]). b. Verify that ( f(x) ) is differentiable on the open interval ((a, b)). c. Check if ( f(a) = f(b) ).

For the Mean Value Theorem: a. Verify that ( f(x) ) is continuous on the closed interval ([a, b]). b. Verify that ( f(x) ) is differentiable on the open interval ((a, b)).
To find the value of ( c ) that satisfies the conclusion of these theorems, follow these steps:

For Rolle's Theorem:
 If ( f(a) = f(b) ), then there exists ( c ) in ((a, b)) such that ( f'(c) = 0 ).

For the Mean Value Theorem:
 The Mean Value Theorem states that there exists ( c ) in ((a, b)) such that ( f'(c) = \frac{{f(b)  f(a)}}{{b  a}} ).
For the function ( f(x) = x^3 + x^2 ), the next steps are to:
 Find the derivative ( f'(x) ).
 Determine the interval ([a, b]) over which you want to apply the theorems.
 Check if the conditions for Rolle's Theorem and the Mean Value Theorem are satisfied on that interval.
 If the conditions are met, solve for ( c ) using the respective conclusions of Rolle's Theorem and the Mean Value Theorem.
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To check the hypotheses of Rolle's Theorem and the Mean Value Theorem for the function ( f(x) = x^3 + x^2 ), and find a value of ( c ) that makes the appropriate conclusion true, follow these steps:

Check the hypotheses of Rolle's Theorem:
 Verify that ( f(x) ) is continuous on the closed interval ([a, b]).
 Verify that ( f(x) ) is differentiable on the open interval ((a, b)).
 Ensure that ( f(a) = f(b) ).

Check the hypotheses of the Mean Value Theorem:
 Verify that ( f(x) ) is continuous on the closed interval ([a, b]).
 Verify that ( f(x) ) is differentiable on the open interval ((a, b)).

Find a value of ( c ) that satisfies the conclusion of Rolle's Theorem:
 If the hypotheses of Rolle's Theorem are met, there exists a ( c ) in ((a, b)) such that ( f'(c) = 0 ).
 Compute the derivative ( f'(x) ) and find its zeros to determine the critical points.
 Check if there exists a ( c ) in ((a, b)) such that ( f'(c) = 0 ).

Find a value of ( c ) that satisfies the conclusion of the Mean Value Theorem:
 If the hypotheses of the Mean Value Theorem are met, there exists a ( c ) in ((a, b)) such that ( f'(c) = \frac{f(b)  f(a)}{b  a} ).
 Compute ( \frac{f(b)  f(a)}{b  a} ) and check if there exists a ( c ) in ((a, b)) such that ( f'(c) = \frac{f(b)  f(a)}{b  a} ).
By following these steps, you can check the hypotheses of Rolle's Theorem and the Mean Value Theorem for ( f(x) = x^3 + x^2 ) and find a value of ( c ) that satisfies the appropriate conclusion.
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