Math Help - Finding a and b (derivatives)

1. Finding a and b (derivatives)

What are the values that a and b can take so that f(x)= ax^2 + bx -sin3x has a local maximum when x=0 ?

2. Originally Posted by sebasto
What are the values that a and b can take so that f(x)= ax^2 + bx -sin3x has a local maximum when x=0 ?

To have a local max at zero

$f'(0)=0$ and $f''(0) < 0$

Using these two gives

$f'(x)=2ax+b-3\cos(3x) \implies f(0)=0=b -3 \implies b=3$

$f''(x)=2a+9\sin(3x) \implies f''(0)=2a \implies 2a < 0 \iff a < 0$

So this function has a max at zero anytime $a < 0 \mbox{ and } b=3$

Here is a graph with a=-1 (black),-3 (red) ,-5 (blue)