Calculus

[markosheehan]

let f(x)=ax³+bx²+cx+d. if b²=3ac show that f(x) has only one turning point. how do i find this

[shyamjayakannan]

Having only 1 turning point means there is only 1 point where f'(x)=0. Now, [tex]f'(x)=3ax^2+2bx+c[/tex]
So, if f'(x) has to have only 1 root, that means the discriminant of the quadratic must be 0. So, we need to check if [tex](2b)^2-4\times3a\times c=0[/tex]
It is given to us that [tex]b^2=3ac[/tex]. Substituting [tex]b^{2 }[/tex] into the above equation, we get 0. Hence, Proved.