Use a graphing calculator such as TI-84 plus 1.

Analyze the polynomial function f(x) = 49x - xAnalyze the polynomial function f(x) = 5x *+ 10x" - 45x 3 - 90xProblem 1: Analyze the polynomial function f(x) = 49z =% Step 1: Find the first derivative. () = (49z 2*) = 49 32? Step 2: Set the first derivative equal to zero to find critical points. 49 - 322 =032 =492 = W /9 4 /B _ 4 VIIT u 4416 (rounded to the nearest hundredth) Step 3: Plug the critical paints back into the original function to find the y-coordinates of the turning points. Forz ~ 4.16: f(4.16) = 49(4.16) (4.16)% ~ 203.84 - 72.01 ~ 131.83 Fora ~ 4.16: f(4.16) = 49( 4.16) (4.16)3 ~ 203.84 + 72.01 ~ 131.83 Turning Points: The turning points are approximately at (4.16, 131.83) and (4.16, 131.83). Problem 2: Analyze the polynomial function f(z) = 5z* + 102* 4522 90z. Step 1: Find the first derivative. f(z) = (52" + 102 452 90z) = 20z* | 30z* 90z 90 Step 2: Set the first derivative equal to zero to find critical points. 202 + 302% 90z 90 =0 We solve this cubic equation for & either numerically or by using a method like the Rational Root Theorem. After solving the cubic equation, we get the approximate roots: e I~ 3 e xv =~ 1.67 Step 3: Plug the critical points back into the original function to find the y-coordinates of the turning points. For a = -3: f(-3) = 5(-3) + 10(-3)3 - 45(-3)2 -90(-3) = 5(81) - 10(27) - 45(9) + 270 = 405 - 270 - 405 + 270 = 0 For x ~ -1.67: f (-1.67) ~ 5(-1.67)4 + 10(-1.67)3 - 45(-1.67)2 -90(-1.67) ~ 39.31 - 46.42 + 125.57 + 150.3 = 268.76 For a = 1: f (1) = 5(1)4 + 10(1)3 - 45(1)2 - 90(1) = 5 + 10 - 45 -90 = -120 Turning Points: The turning points are approximately at (-3, 0), (-1.67, 268.76), and (1, -120). Summary: 1. For the first function f (x ) = 49x - a : The turning points are approximately at (4.16, 131.83) and (-4.16, -131.83). 2. For the second function f(x) = 5x4 + 10x3 - 45x2 - 90x: The turning points are approximately at (-3, 0), (-1.67, 268.76), and (1, -120)