Find the range of $\displaystyle \frac{1}{2bx - (x^2 + b^2 + \sin^2 x)}$, $\displaystyle x\in [-1,0], b\in [2,3]$.
Generaly:
Bol. - Weier. theorem says: if f is continuous on [ a, b ] then f( [ a, b ] ) = [ c, d ] segment. Candidates for min and max ( c and d ) are:
1. z = a and z = b ( f( a ) and f( b ) )
2. f is differential in z and f ' ( z ) = 0 ( f( z ) )
3. f is not differential in z