\(y=x^5-3\\ y'=5x^4\\\\ 5x^4=0\\ x=0\\ 0\in [-2,1]\\\\ y''=20x^3\\\\
y''(0)=20\cdot0^3=0\)
The value of the second derivative for \(x=0\) is neither positive nor negative, so you can't tell whether this point is a minimum or a maximum. You need to check the values of the first derivative around the point.
But the value of \(5x^4\) is always positive for \(x\in\mathbb{R}\setminus \{0\}\). That means at \(x=0\) there's neither minimum nor maximum.
The maximum must be then at either of the endpoints of the interval \([-2,1]\).
The function \(y\) is increasing in its entire domain, so the maximum value is at the right endpoint of the interval.
\(y_{max}=y(1)=1^5-3=-2\)
