The graph of {{MATH_247}} (the derivative of {{MATH_248}} ) is shown below. Assume {{MATH_249}} is continuous for all real numbers. {{CMD_250}} {{ENV_251}} {{CMD_252}} {{CMD_253}} On which of the following intervals is {{MATH_254}} increasing? {{CMD_255}} {{CMD_256}}{{MATH_257}}{{CMD_258}} {{CMD_259}}{{MATH_260}}{{CMD_261}} {{CMD_262}}{{MATH_263}}{{CMD_264}} {{CMD_265}}{{MATH_266}}{{CMD_267}} {{CMD_268}}{{MATH_269}}{{CMD_270}} {{CMD_271}} {{CMD_272}} {{MATH_273}} is increasing where {{MATH_274}}, i.e. on the intervals {{MATH_275}} and {{MATH_276}}. {{CMD_277}} {{CMD_278}} {{CMD_279}} Which of the following are critical points of {{MATH_280}}? {{CMD_281}} {{CMD_282}}{{MATH_283}}{{CMD_284}} {{CMD_285}}{{MATH_286}}{{CMD_287}} {{CMD_288}}{{MATH_289}}{{CMD_290}} {{CMD_291}}{{MATH_292}}{{CMD_293}} {{CMD_294}} {{CMD_295}} {{MATH_296}} has a critical point at the zeros of {{MATH_297}}, and the places where {{MATH_298}} does not exist. In this case, {{MATH_299}}, {{MATH_300}}, and {{MATH_301}}. {{CMD_302}} {{CMD_303}} {{CMD_304}} Where do the local maxima occur? {{CMD_305}} {{CMD_306}}{{MATH_307}}{{CMD_308}} {{CMD_309}}{{MATH_310}}{{CMD_311}} {{CMD_312}}{{MATH_313}}{{CMD_314}} {{CMD_315}}{{MATH_316}}{{CMD_317}} {{CMD_318}} {{CMD_319}} A local maximum occurs at a critical point where the function transitions from increasing to decreasing, i.e. the derivative passes from positive to negative. In this case, we see that the local maxima occur at {{MATH_320}} and {{MATH_321}}. {{CMD_322}} {{CMD_323}} {{CMD_324}} Where does a point of inflection occur? {{CMD_325}} {{CMD_326}}{{MATH_327}}{{CMD_328}} {{CMD_329}}{{MATH_330}}{{CMD_331}} {{CMD_332}}{{MATH_333}}{{CMD_334}} {{CMD_335}}{{MATH_336}}{{CMD_337}} {{CMD_338}} {{CMD_339}} A point of inflection occurs when the concavity of {{MATH_340}} changes. This is reflected in the sign of {{MATH_341}} changing. This only occurs at one point in this graph, namely {{MATH_342}}. {{CMD_343}} {{CMD_344}}