Please answer the following questions about the function If you are asked to fin
ID: 2872902 • Letter: P
Question
Please answer the following questions about the function If you are asked to find x or y values enter either a number a list of numbers separated by commas, or none if there aren't any solutions. Use interval notation if you are asked to find an interval or union of intervals and either if the interval is empty Find the critical numbers of f, where it is increasing and decreasing and its local extrema Find where f is concave up, concave down and has inflection points Find any horizontal and vertical asymptotes of f
Explanation / Answer
a)f'(x)=3*5x4-2*4x3=0
==>15x4-8x3=0
==>x3(15x-8)=0
==>x=0, 15x-8=0
==>x=0,x=8/15
increasing ==>f'(x)>0 ==>x3(15x-8)>0==>(-inf,0)U(8/15, inf)
decreasing ==>f'(x)<0 ==>x3(15x-8)<0==>(0,8/15)
f"(x)=15*4x3-8*3x2=60x3-24x2
f"(0)=0==>local maxima
f"(8/15)=60(8/15)3-24(8/15)2=32(8/15)2 >0==> local minima at x=8/15
b)
inflection points
f"(x)=60x3-24x2=0==>12x2(5x -2)=0==>x=0,x=2/5
f "(-1/5)<0,f "(1/5)<0 ==>x=0 is not inflection point
f "(1/5)<0,f "(3/5)>0==>x=0 is inflection point
x=2/5=0.4
concave up
f"(x)=60x3-24x2>0==>12x2(5x -2)>0==>x>2/5==>x>0.4
x=(0.4,inf)
concave down
f"(x)=60x3-24x2<0==>12x2(5x -2)<0==>x<2/5==>x<0.4
x=(-inf,0.4)
c)no horizontal or vertical asymptotes
d)f is continous because it is defined for all x in domain of f.
no symmetry