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PT 4 solutions

# PT 4 solutions - PRACTICE TEST 4 SOLUTIONS 1 Find intervals...

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PRACTICE TEST 4 SOLUTIONS 1. Find intervals on which f is increasing, decreasing; open intervals where f is concave up, concave down; and x coordinates of all inflection points; also state which points are stationary points, PONDS, and classify as relative maximum or minimum points. ( 2 3 2 2 f x x x x = - + - IT’S A POLYNOMIAL – SO NO PONDS! ( ( ( ( 2 2 1 4 3 3 4 1 3 1 1 f x x x x x x x = - + - = - - + = - - - so there are 2 stationary points, at x = 1 and x = 1/3 Sign graph shows f < 0 for x < 1/3 and x > 1; f > 0 for 1/3 < x < 1; hence there’s a rel min at x = 1/3 and rel max at x = 1 ( ( 4 6 6 2/3 f x x x ′′ = - = - 0 f ′′ if x < 2/3, f < 0 for x > 2/3 and an inflection point at x = 2/3 2. GRAPH Find and show intercepts vertical and horizontal asymptotes and SHOW SIGN GRAPHS FOR f AND f ′′ : ( 29 2 4 3 x f x x - = - GIVEN: ( 29 ( 29 2 2 ; 3 f x x = - ( 29 ( 29 3 4 3 f x x ′′ = - (2, 0) and (0, - 4/3) are intercepts; x = 3 is a vertical asymptote and y = -2 is a horizontal asymptote; f > 0 [except at x = 3], so f is always increasing [and no sign graph needed for f ] 0 f ′′< if x < 3 [hence f concave down]; 0 f ′′ if x > 3 [hence f concave up] Note: the vertical asmyptote should be drawn as a broken line – but I can’t show it here. 3. Find the absolute maximum and minimum values of ( 29 [ ] 1 sin on 0, 2 f x x x π = + Since this is on a closed interval, the Extreme Value Theorem lets us find critical points then just evaluate f at the end points and critical point[s]

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