Question

1. Give a complete list of all numbers a for which z2 +1 > ar. 2. Definition: A function f is even if f(-x) = f(x) for all inputs z. A function f is odd if f(-x) = -f(x). (a) Let f be any function with domain (-0,0). i. Show that the function g(x) = f(x) + f(-x) is even. ii. Show that the function h(x) = f(0) - f(-x) is odd. iii. Show that f can be written as a sum of an even and odd function. (b) Show that any differentiable odd function is even. (c) Show that any differentiable even function is odd. 3. Let f(x) = (x – 6). Show that f and f' share a root. 4. Let f(x) = x2 + ax + b. Suppose f and f' share a root r. Show that f(x) = (x – r)?

Answer 1

1. x² + 7 ax => ax < x² + => a {xth => a nxt (if x 20) rit xco) 9(-x) = f(-x) + f(-1-x)) = f(-x) + f(x) = g(x) g is even. h(-x) = f(-x) - f (x) -- [ f(x) - f(-x)). (11) - h (a) = h is odd. (11%) since 2 f(x) = g(x) + h(x) . f(x) = 1 g(x) + 2 h (x) hence t can be written as sum of even and odol.

det f is an odd Junction then fix = lim flath)-f(x) f'(-x) = hin f(-2 +h)- f(-x) 450 ch . as fis odd = lim -f(x-4) + f(x) 490 % - ( few.h2 times) En liten fluth)-f(0) -:- (-f'(x)) = f'(x) a f is even. - © eit t = f is even dim t (ath)-f(x) f'(-x) = limf. (anth)-f(-4) hto a lim f(-(8-41)-f(-x) nano

lim trech)-f(x) no = -f(x) =) is odd. f(x)= (x-b)? Here aab is root of f(x) because t (6)= (b-b]²=0 Also f'(x) = 2 (x-b] x=b is also root of flex) pl(b) = 2 (b-b) - 0 a fit shares root. 1. x) f(x)=x²tax tb m is a root q = r2 tartboo is also root of s flex)= 2x ta fix) 27 ta = 0 을

Then (E)-2.a +6=0 - a² + b = 0 472 2 rence 2 flu) = x. narx tra = (x-V)?