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 ?
Thanks in advance
To have a local max at zero
$\displaystyle f'(0)=0$ and $\displaystyle f''(0) < 0$
Using these two gives
$\displaystyle f'(x)=2ax+b-3\cos(3x) \implies f(0)=0=b -3 \implies b=3$
$\displaystyle f''(x)=2a+9\sin(3x) \implies f''(0)=2a \implies 2a < 0 \iff a < 0$
So this function has a max at zero anytime $\displaystyle a < 0 \mbox{ and } b=3$
Here is a graph with a=-1 (black),-3 (red) ,-5 (blue)