Suppose that {{MATH_140}} is the linear approximation of a differentiable function {{MATH_141}} at {{MATH_142}}. Suppose {{MATH_143}} is a different {{MATH_144}}-value near {{MATH_145}} and {{MATH_146}} is the open interval between {{MATH_147}} and {{MATH_148}}. Then: {{CMD_149}} {{CMD_150}} If {{MATH_151}} is concave up on the interval {{MATH_152}}, then the approximation value {{MATH_153}} is an {{CMD_154}}underestimate{{CMD_155}} of {{MATH_156}}. {{CMD_157}} If {{MATH_158}} is concave down on the interval {{MATH_159}}, then the approximation value {{MATH_160}} is an {{CMD_161}}overestimate{{CMD_162}} of {{MATH_163}}. {{CMD_164}} {{CMD_165}} {{CMD_166}} To determine whether {{MATH_167}} is an overestimate or an underestimate, it is not enough to check the value of {{MATH_168}} at a single point. It's the concavity over the entire interval between the points {{MATH_169}} and {{MATH_170}} that has to be checked. {{CMD_171}}