Sketch the graph of a twice-differentiable function `y=f(x)` whose derivatives have the following sign patterns below. Also you are given:`f(-4)=f(-1)=0, \text{ }f'(-4)=f'(-2)=0,\text{ }f''(-3)=0`.
Interval
`(-oo,-4)`
`(-4,-2)`
`(-2,oo)`
`f'(x)`
`\text{ }+`
`\text{ }-`
`\text{ }+`
Interval
`(-oo,-3)`
`(-3,oo)`
`f''(x)`
`\text{ }-`
`\text{ }+`
To sketch this function, focus on the key features provided by the sign patterns of `f'(x`) and `f''(x)`, as well as the given points where `f(x) = 0`, `f'(x) = 0`, and `f''(x) = 0`. Start by plotting these critical points. Then, use the sign of `f'(x)` to determine where the function is increasing or decreasing, and the sign of `f''(x)` to determine the concavity. Remember that a change in the sign of `f'(x)` indicates a local extremum, while a change in the sign of `f''(x) ` indicates an inflection point.