3.4 Practice

Question 1

Determine the open intervals on which the graph of y = –3x^3 + 8x^2 + 6x – 8 is concave downward or concave upward.

Answer 1

concave upward on ( -∞, 8/9 )

concave downward on ( 8/9, ∞ )

Question 2

Determine the open intervals on which the graph of the function f(x) = (x^2) / (x^2 + 64) is concave upward or concave downward.

Answer 2

concave upward: ( -(8√3) / 3, (8√3) / 3 )

concave downward: ( -∞, -(8√3) / 3 ) and ( (8√3) / 3, ∞)

Question 3

Find the points of inflection and discuss the concavity of the function f(x) = x √(x+14).

Answer 3

no inflection points; concave up on (–14, ∞)

Question 4

Find the points of inflection and discuss the concavity of the function f (x) = –3x + cosx on the interval [0,2π ] .

Answer 4

concave upward on ( π/2, 3π/2)
concave downward on ( 0, π/2) and (3π/2, 2π)
inflection points ( π/2, -3π/2) and (3π/2, -9π/2)

Question 5

If f(3) = 8, f’(3) = 0, and f’‘(3) = 7, what (if anything) can I say about the graph of f at x = 3?

Answer 5

(3, 8) is a relative minimum

Question 6

If f(1) = 4, f’(1) = 0, and f’‘(1) = -3, what (if anything) can I say about the graph of f at x = 1?

Answer 6

(1, 4) is a relative maximum

Question 7

If f(4) = 1, f’(4) = 0, and f’‘(4) = 0, what (if anything) can I say about the graph of f at x = 4?

Answer 7

can ONLY say that there is a horizontal tangent line at (4, 1)