Question: Which of the following statement(s) is/are true for the graph of the function f(x) = x4 - 4x2 + 4? I. The graph is concave upward in (- ∞, - 2). II. x - intercept is (4, 0). III. y - intercept is (0, 4). IV. The graph is increasing in (2, ∞). V. Critical numbers are x = 0, - 2, 2.
A ) II only B ) III only C ) III and V only D ) I, III and IV only E ) II and IV only
Steps to derive
1f(x) = x4 - 4x2 + 4 [Write the function.]
2x4 - 4x2 + 4
= 0 ⇒x = ± 2 x - intercepts are (- 2, 0), (2, 0) [Equate f(x) to zero to find x-intercepts and solve for x.]
3f(0) = 4, so y - intercept is (0, 4) [Put x = 0 in f(x) to find y - intercept.]
4f ′(x) = 4x3-8x [Find f ′(x).]
5 4x3-8x = 0 ⇒x = 0, - 2, 2 Critical numbers are x = 0, x = - 2, x = 2 [Equate f ′(x) to zero to find critical numbers.]
6f ″(x) = 12x2-8 [Find f ″(x).]
7f ″(- 2) = 16 > 0, f ″(0) = - 8 < 0, f ″(2) = 16 > 0
8 Draw the graph of the function f(x).
9 The graph of f(x) is : decreasing and concave upward in (- ∞, - 2), increasing and concave upward in (- 2, - 23), increasing and concave downward in (- 23, 0), decreasing and concave downward in (0, 23), decreasing and concave upward in (23, 2), increasing and concave upward in (2, ∞). [f ″(x) = 0 at x = ± 23.]
