\fAnswer: Consider the given graph: d2y The graph near point a is concave up, hence 0 , thus it cannot be inflection point. dx 2 Similarly the graph near point b is concave up, hence d2y 0 , thus it cannot be dx 2 inflection point. 2 d y Next the graph near point c is concave down; hence 2 0 , thus it cannot be inflection dx point. Although at point d , graph is changing concavity from negative to positive but the function has no derivative at point d (graph has corner at point d , hence not smooth), hence it cannot be inflection point. However at point e , graph is changing concavity from positive to negative also graph is differentiable (smooth) at point e , therefore graph has inflection point at point e . Next the graph near point f is concave down; hence point. d2y 0 , thus it cannot be inflection dx 2 Answer: Consider the given graph: The graph near point a is concave up, hence d2y 0 , thus it cannot be inflection point. dx 2 Similarly the graph near point b is concave up, hence d2y 0 , thus it cannot be dx 2 inflection point. Next the graph near point c is concave down; hence d2y 0 , thus it cannot be inflection dx 2 point. Although at point d , graph is changing concavity from negative to positive but the function has no derivative at point d (graph has corner at point d , hence not smooth), hence it cannot be inflection point. However at point e , graph is changing concavity from positive to negative also graph is differentiable (smooth) at point e , therefore graph has inflection point at point e . Next the graph near point f is concave down; hence point. d2y 0 , thus it cannot be inflection dx 2 Answer: Consider the given graph: d2y The graph near point a is concave up, hence 0 , thus it cannot be inflection point. dx 2 Similarly the graph near point b is concave up, hence d2y 0 , thus it cannot be dx 2 inflection point. 2 d y Next the graph near point c is concave down; hence 2 0 , thus it cannot be inflection dx point. Although at point d , graph is changing concavity from negative to positive but the function has no derivative at point d (graph has corner at point d , hence not smooth), hence it cannot be inflection point. However at point e , graph is changing concavity from positive to negative also graph is differentiable (smooth) at point e , therefore graph has inflection point at point e . Next the graph near point f is concave down; hence point. d2y 0 , thus it cannot be inflection dx 2 Answer: Consider the given graph: The graph near point a is concave up, hence d2y 0 , thus it cannot be inflection point. dx 2 Similarly the graph near point b is concave up, hence d2y 0 , thus it cannot be dx 2 inflection point. Next the graph near point c is concave down; hence d2y 0 , thus it cannot be inflection dx 2 point. Although at point d , graph is changing concavity from negative to positive but the function has no derivative at point d (graph has corner at point d , hence not smooth), hence it cannot be inflection point. However at point e , graph is changing concavity from positive to negative also graph is differentiable (smooth) at point e , therefore graph has inflection point at point e . Next the graph near point f is concave down; hence point. d2y 0 , thus it cannot be inflection dx 2