Let a, 3 are real numbers, 3> 0, and o is defined on [-1,1] by 2...

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Let a, 3 are real numbers, 3> 0, and o is defined on [-1,1] by 2" cos(a|-), x#0 r = 0 O(x) = 0, Prove the followings i. o is continues if and only if a > 0. ii. o (0) exists if and only if a > 1. iii. o' is bounded if and only if a >1+B. iv. o' is continues if and only if a > 1+3. v. o"(0) exists if and only if a > 2+ B. vi. " is bounded if and only if a 2 2+ 23. vii. ø" is continuous if and only if a > 2+ 23. Let a, 3 are real numbers, 3> 0, and o is defined on [-1,1] by 2" cos(a|-), x#0 r = 0 O(x) = 0, Prove the followings i. o is continues if and only if a > 0. ii. o (0) exists if and only if a > 1. iii. o' is bounded if and only if a >1+B. iv. o' is continues if and only if a > 1+3. v. o"(0) exists if and only if a > 2+ B. vi. " is bounded if and only if a 2 2+ 23. vii. ø" is continuous if and only if a > 2+ 23.