For which values of k will x^3-x^2-8x+k=0 have three real roots
Let f(x)=x^3-x^2-8x=x(x^2-x-8).
f(x) has 3 real roots.
f'(x)=3x^2-2x-8=(3x+4)(x-2)
maxf(x)=f(-4/3)=-64/27-16/9+32/3=176/27
minf(x)=f(2)=8-4-16=-12
-176/27<k<12
f(x) has 3 real roots.
f'(x)=3x^2-2x-8=(3x+4)(x-2)
maxf(x)=f(-4/3)=-64/27-16/9+32/3=176/27
minf(x)=f(2)=8-4-16=-12
-176/27<k<12