Derivative and extremum

Question

I try to solve a problem but I just can't figure it out:

I have this equation as a statement:

$f(x)= x^4+ax^3+bx^2+cx+d$

There is a maximum at $x=0$

There are four propositions and one is supposed to be true (apparently the second one):

• $d=0$ and $b>0$
• $c=0$ and $b<0$
• $c=0$ and $b>0$
• $d=0$ and $b<0$

I get why d doesn't matter as the first derivative of the function will make it disappear anyway.

I get why c should equal to $0$ because it needs to be so that the first derivative equals $0$ when $x=0$.

But I really don't get why b should necessarily be $<0$ to make this max possible for this $f(x)$ equation.

Any help would be appreciated.

Thanks.

Answer 1

Calculate the second derivative at $0$. To get a maximum, $f''(0)$ has to be negative. If $f''(0)$ is positive, then you have a minimum.