Determine the open intervals on which the graph of f(x) = 3x - 2 cos x is concav
ID: 2893244 • Letter: D
Question
Determine the open intervals on which the graph of f(x) = 3x - 2 cos x is concave downward or concave upward. A. concave upward on .., (-3 pi/6, -pi/6), (pi/6, 3 pi/6), (5 pi/6, 7 pi/6), ..: concave downward on .., (-5 pi/6, 3 pi/6), (-pi/6, pi/6), (3 pi/6, 5 pi/6), ... B. concave downward on .., (-3 pi/4, -pi/4), (pi/4, 3 pi/4), (5 pi/4, 7 pi/4), ...: concave downward on .., (-5 pi/4, 3 pi/4), (-pi/4, pi/4), (3 pi/4, 5 pi/4), ... C. concave upward on .., (-3 pi/2, -pi/2), (pi/2, 3 pi/2), (5 pi/2, 7 pi/2), ....: concave downward on ..., (-5 pi/2, -3 pi/2), (-pi/2, pi/2), (3 pi/2, 5 pi/2), .... D. concave downward on .., (-3 pi/2, -pi/2), (pi/2, 3 pi/2), (5 pi/2, 7 pi/2), ...: concave upward on .., (-5 pi/2, 3 pi/2), (-pi/2, pi/2), (3 pi/2, 5 pi/2), ... Determine the open intervals on which the graph of y = -x^3 + 2x^2 + 3x - 2 is concave downward or concave upward. A. concave downward on (-infinity, infinity) B concave downward on (-infinity, 2/3): concave upward on (2/3, infinity) C. concave downward on (-infinity, 2/3): concave upward on (-2/3, infinity) D. concave upward on (-infinity, 2/3): concave downward on (-infinity, 2/3): concave downward on (2/3, infinity) E. concave upward on (-infinity, 2/3): concave downward on (-2/3, infinity)Explanation / Answer
interval for which
f''(x) > 0 then f(x) concave upwards,
f''(x) < 0, then f(x) concave downwards
13.
f(x) = 3x - 2*cos x
f'(x) = 3*1 - 2*(-sin x)
f'(x) = 3 + 2*sin x
f''(x) = 0 + 2*cos x
2*cos x > 0, in interval [-4*pi, 4*pi]
when x = (-4pi, -7pi/2) U (-5pi/2, -3pi/2) U (-pi/2, pi/2) U (3pi/2, 5pi/2) U (7pi/2, 4pi)
2*cos x < 0, in interval [-4*pi, 4*pi]
when x = (-7pi/2, -5pi/2) U (-3pi/2, -pi/2) U (pi/2, 3pi/2) U (5pi/2, 7pi/2)
from above intervals, correct option is D.
14.
f(x) = -x^3 + 2x^2 + 3x - 2
f'(x) = -3x^2 + 4x + 3
f''(x) = -6x + 4
-6x + 4 > 0, when x < 2/3
-6x + 4 < 0, when x > 2/3
Correct option is D.
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