5. (4 pts) Fill in the blank in each sentence below to describe a feature on a slope graph given a feature on its corresponding function graph. Choose from: "be increasing", "be decreasing", "lie above the x -axis", or "lie below the $$ x $$-axis", "zero", "positive", "negative", "relative max or min", "inflection point", "vertical asymptote", "herizontal asymptote". "bole", "jump" a) If a continuous and differentiable function $$ f(x) $$ has a relative minimum at $$ x=1 $$, then its slope graph must have a/an $$ \qquad $$ at $$ x=1 $$. b) If a continuous and differentiable function $$ g(x) $$ has an inflection point at $$ x=-2 $$, then its slope graph must have a/an $$ \qquad $$ at $$ x=-2 $$. c) If a continuous function $$ h(x) $$ has a vertical tangent at $$ x=0 $$, then its slope graph has $$ \qquad $$ at $$ x=0 $$. d) If a continuous and differentiable function $$ j(x) $$ is decreasing on the interval $$ (0,10) $$, then its slope graph must $$ \qquad $$ on the interval $$ (0,10) $$. e) If a continuous and differentiable function $$ j(x) $$ is concave up on the interval $$ (0,10) $$, then its slope graph must $$ \qquad $$ on the interval $$ (0,10) $$.

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a) “zero”  
 At a relative minimum for f, f ′(1) = 0 (with f ′ changing from negative to positive).

b) “relative max or min”  
 At an inflection point for g, the slope graph (g ′) has a local extremum because g ″ changes sign.

c) “vertical asymptote”  
 A vertical tangent at x = 0 makes h′ tend to infinity (or minus infinity), so the slope graph has a vertical asymptote there.

d) “lie below the x‑axis”  
 If j is decreasing, then its derivative is negative, so the slope graph lies below the x‑axis on (0, 10).

e) “be increasing”  
 Concave up implies j ″>0, meaning j ′ (the slope graph) is increasing on (0, 10).
a) zero b) relative max or min c) vertical asymptote d) lie below the x-axis e) be increasing

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