A graph of {{MATH_43}} is given below, with domain {{MATH_44}}. {{CMD_45}} {{ENV_46}} {{CMD_47}} On what intervals is {{MATH_48}} concave up? On what intervals is {{MATH_49}} concave down? {{CMD_50}} Let's draw some tangent lines on the graph. {{CMD_51}} {{ENV_52}} {{CMD_53}} The graph is above the tangent lines for {{MATH_54}} in {{MATH_55}}, and below the tangent lines we sketched on {{MATH_56}}. Therefore, {{MATH_57}} is concave up on {{MATH_58}}, and concave down on {{MATH_59}}. {{CMD_60}} {{CMD_61}} In that example, examine the intervals where {{MATH_62}} was concave up. At the beginning of that interval, {{MATH_63}} was decreasing, and at the end, {{MATH_64}} was increasing. The slopes in that interval increased from a negative value to a positive value. In the interval where {{MATH_65}} was concave down, the slopes started positive and ended negative. The slopes in that interval were decreasing.