# 21 and 22 Please dont answer the question if you have horrible handwriting Usi
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# 21 and 22
Please dont answer the question if you have horrible handwriting
Using Rolle's Theorem In Exercises 9-22, determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f'(c)0. If Rolles Theorem cannot be applied, explain why not. 9. f(x) - -x2 + 3x. [0, 3] 10. f(x)-x2 - 8x +5, [2, 6] 12. f(x-(x - 4)(r 13. f(x) = x2/3-1, 2)2, [-2,4] [-8, 8] 14. f(x)-3-1x-31, [0, 6] r22x 15. f(x) = x + 2 17. f(x)-sin x, [0, 2?] 19, f(x) tan x, [0, ?] 21. f(x)-(x2-20e. [0, 2] 18. f(x)-cos 2x, [-?, ?] 20. f(x) 22. f(x)-x-2 In x, [1, 3] T. 2?Explanation / Answer
21. f(x) = (x2-2x)ex
f(0)= (0-0) e0=0
f(2) = (4-4)e2=0
f(0)=f(2)
f'(x) = (2x-2)ex+(x2-2x)ex
= ex(2x-2 +x2-2x) = ex(x2-2)
f '(c) =0
ec(c2-2)=0
c2-2=0
c = sqrt 2 tat belongs to [0,2]
22. f(x) = x -ln x
f(1) = 1-2ln 1 = 1
f(3) = 3-2ln 3
So here f(1) not equals to f(3), so Rolle's theorem cant apply here .
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