Question

For the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither; b) identify intervals where the function is increasing or decreasing; c) give the coordinates of any points of inflection; d) identify intervals where the function is concave up or concave down, and e) sketch the graph. h(x) = x - 24x

Answer 1

Given nix) = 273_24x For Critical points h'(a) = 0 → h'ln) = 6x2-24 = 0 + 60°= 24 = x2 = 4 + 12 = 12 x=-2, 2 are critical points. Now ni (2) = 12x. and M" (-2) = -24 CO = The critical point x=-2 is relative maximum Also (2)=2470 > the critical point a=2 is relatie minimum. a) (b) for fonesion to be increasing we have h'(a) no now new=62²24 = 6n²> 24 = x²x4 = 22-2 and 272 : non is increasing on (-00,-2) 1 and (2,00) nie hen) is increasing on f-00, 2) U (2,00) For - deeneasing we have h (alco + 6x²-24 <0 2) x² <4 » aca and x>-2 head is Decreasing on -2 Laar lie on (-2, 2)

(d) © For inflection points we have h" (=0 & 122=0 =)/2=0 is the inflection point for con came up we have - n1(a) to 12270 => 370 . hon) is concave up on - 10,00) . For concave down we have h(n). Los = 123 Zo = 250 a how is concame down on (-0,0). . .

increasing decreasing • critical point -6 -4 2 4 6 - 100