A. Sketch a graph of the function \( f \) that satisfies all of the following conditions.
(a) \( f(-4)=0 \) and \( f(0)=3 \)
(b) \( \displaystyle\lim_{x\to4^-} f(x) = -\infty \) and \( \displaystyle\lim_{x\to4^+} f(x) = \infty \)
(c) \( \displaystyle\lim_{x\to\infty} f(x) = 2 \)
(d) \( f'(x) > 0 \) on \( (-\infty, 0) \)
(e) \( f'(x) < 0 \) on \( (0,4) \) and \( (4,\infty) \)
B. Sketch a graph of the function \( f \) that satisfies all of the following conditions.
(a) \( f(-2)=0 \), \( f'(-2)=0 \), and \( f(0)=3 \)
(b) \( f'(x) < 0 \) on \((-\infty,-2)\), and \( f'(x) > 0 \) on \((-2,0)\)
(c) \( f''(x) > 0 \) on \((-\infty,-1)\), and \( f''(x) < 0 \) on \((-1,1)\)
(d) The graph has y-axis symmetry.
C. Sketch the graph of the function \( f \) that satisfies the given conditions.
(a) \( f'(-3)=f'(0)=0 \)
(b) \( \displaystyle\lim_{x\to3^+} f(x) = -\infty \),
\( \displaystyle\lim_{x\to3^-} f(x) = \infty \)
(c) \( \displaystyle\lim_{x\to\infty} f(x) = -1 \)
(d) \( f'(x) > 0 \) on \((-\infty,-3)\), \((0,3)\), and \((3,\infty)\)
(e) \( f'(x) < 0 \) on \((-3,0)\)
(f) \( f''(x) > 0 \) on \((-1.5, 3)\)
(g) \( f''(x) < 0 \) on \((-\infty,-1.5)\) and \((3,\infty)\)