Which one of the following statements describes the shape of the graph of the function, f(x)= 2x5 - 3x4 + 2x2 - 17x + 31, at the point, x = - 1?
A)Tangent slopes upwards and graph bends downwards
B)Tangent slopes downwards and graph bends downwards
C)Tangent slopes downwards and graph bends upwards
D)Tangent slopes upwards and graph bends upwards
E)Tangent is horizontal

Question

Quizplus · Accepted Answer

The derivative of the function is f'(x) = 10x^4 - 12x^3 + 4x - 17. At x = -1, f'(-1) = 10(-1)^4 - 12(-1)^3 + 4(-1) - 17 = 10 - 12 - 4 - 17 = -19. Since f'(-1) < 0, the tangent line slopes downwards. The second derivative of the function is f''(x) = 40x^3 - 36x^2 + 4. At x = -1, f''(-1) = 40(-1)^3 - 36(-1)^2 + 4 = -40 - 36 + 4 = -72. Since f''(-1) < 0, the graph bends downwards.

Which One of the Following Statements Describes the Shape of the Graph