(a) Use properties of quadratic functions to prove that (x 1)2 2. (c) Let g(n)...

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(a) Use properties of quadratic functions to prove that (x 1)2 <3x² for all real x 2. (b) Use mathematical induction and the inequality from part (a) to prove that 5+1 + (n-1)²< 6+1+6 for all integers n > 2. (c) Let g(n) = 5+1 + (n-1)² and h(n) = 6". Using the inequality from part (b), prove that g(n) = O(h(n)). You need to give a rigorous proof derived directly from the definition of the O- notation, without using any theorems from class. (First, give a complete statement of the definition. Next, show how g(n) = O(h(n)) follows from this definition.) (a) Use properties of quadratic functions to prove that (x 1)2 <3x² for all real x 2. (b) Use mathematical induction and the inequality from part (a) to prove that 5+1 + (n-1)²< 6+1+6 for all integers n > 2. (c) Let g(n) = 5+1 + (n-1)² and h(n) = 6". Using the inequality from part (b), prove that g(n) = O(h(n)). You need to give a rigorous proof derived directly from the definition of the O- notation, without using any theorems from class. (First, give a complete statement of the definition. Next, show how g(n) = O(h(n)) follows from this definition.)