Find the absolute extreme values for fx x 327x on the inter

Find the absolute extreme values for f(x) = x 3-27x on the interval [0,4]

Solution

1)

given f(x)=x³-27x

differentiate with respect to x

f \'(x)=3x²-27

for critical numbers , f \'(x)=0

=>3x²-27=0

=>x²=9

=>x=3

f(0)=0

f(3)=3³-(27*3) =-54

f(4)=4³-(27*4) =-44

absolute maximum =0

absolute minimum =-54

---------------------------------------------------------------------------

2)

given y\'=3x²+2x-1

y=y\'dx

y=[3x²+2x-1]dx

y=(3_/3)x³+(2_/2)x²-x+C

y=x³+x²-x+C

----------------------------------------------------------

given y\'=(1_/x²)+2x

y=((1_/x²)+2x)dx

y=((x^-2)+2x)dx

y=(1_/-1)x^-2+1+(2_/2)x²+C

y=-x^-1+x²+C

y=-(1_/x)+x²+C

given point (-1,1).

=>x=-1, y=1

=>1=-(1_/-1)+(-1)²+C

=>1=1+1+C

=>C=-1

so the function is y=-(1_/x)+x²-1