Curve Sketching:
A. Sketch a graph of the function \( f \) that satisfies all of the following conditions. Label all local max/min, asymptotes etc, if any.
(a) \(f(-4)=0\) and \(f(0)=3,\) (b) \(\displaystyle {\lim_{x \to 4^-} f(x) = -\infty}\) and \(\displaystyle{\lim_{x \to 4^+} f(x) = \infty,}\) (c) \(\displaystyle {\lim_{x \to \infty} f(x) =2,}\) (d) \(f'(x) >0 \) on the interval \( (-\infty, 0),\) (e) \( f'(x) < 0 \) on the intervals \( (0,4)\), and \( (4,\infty)\).
B. Sketch a graph of the function \( f \) that satisfies all of the following conditions. Label all local max/min, asymptotes etc, if any.
(a) \( f(-2)=0 \), \( f'(-2)=0\), and \( f(0)=3\)
(b) \( f'(x) <0 \) on the interval \((-\infty,-2)\), and \( f'(x) >0 \) on \((-2,0)\)
(c) \( f''(x)>0 \) on the interval \( (-\infty, -1)\), 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. Label all local max/min, asymptotes etc, if any.
(a) \( f'(-3)=f'(0)=0\)
(b) \(\displaystyle{\lim_{x \to 3^+} f(x)= -\infty}\) , \( \displaystyle{\lim_{x \to 3^-} f(x)=\infty}\)
(c) \( \displaystyle{\lim_{x \to \infty} f(x) = -1}\)
(d) \( f'(x) > 0\) on the intervals \((-\infty,-3), \, (0,3)\) and \((3,\infty)\)
(e) \( f'(x) < 0\) on the interval \( (-3,0)\)
(f) \( f''(x) > 0\) on the interval \((-1.5, 3)\)
(g) \(f''(x) < 0\) on the intervals \((-\infty,-1.5)\) and \((3,\infty)\).