Let's mimimize

Calculus Level 4

If the curve $f(x)=2{ x }^{ 3 }+a{ x }^{ 2 }+bx$ , where $a$ and $b$ are positive integers , cuts the $x$ -axis at three distinct points. Then find the minimum value of $a +b$ .

Bonus: Also find all $f(x)$ for which $a+b$ is minimum.

The answer is 4.

1 solution

Rishabh Jain
Jun 30, 2016

$f(x)=x(2x^2+ax+b)$

Obviously $f(x)$ intersects x-axis at $x=0$ and acc to ques then the quadratic inside brackets must have two distinct roots( i.e discriminant $>0$ ) other than $0$ (i.e value of quadratic at $x=0$ must be non zero which is true since $b\in N$ ).

$a^2-8b>0$

The minimum value of $a+b$ satisfying the above relation along with being natural occurs when $(a,b)\equiv (3,1)$ . Hence, $a+b=\boxed 4$ .

Substituting values of $a,b$ in $f(x)$ we get:-

$f(x)=2x^2+3x^2+x$

Let's mimimize

The answer is 4.

1 solution

0 pending reports