12.43 $f(x)=2^{\sqrt[3]{x^{2}}}:$ а) on $[-8,-1]$; б) on $[-1,1]$.
12.43 Find \( f^{\prime}(x) = 2^{\sqrt[3]{x^{2}}} \ln 2 \cdot \frac{2}{3} x^{-1 / 3} \); \( f^{\prime}(x) \neq 0 \) for any \( x \), \( f^{\prime}(x) \) does not exist at \( x = 0 \).

a) If \( x \in [-8, -1] \), then \( f(-8) = 2^{4} = 16 \), \( f(-1) = 2 \);
thus, \( y_{\text{min}} = 2 \), \( y_{\text{max}} = 16 \).

b) If \( x \in [-1, 1] \), then \( f(-1) = 2 \), \( f(1) = 2 \), \( f(0) = 1 \), hence \( y_{\text{min}} = 1 \), \( y_{\text{max}} = 2 \).

Answer: a) \( y_{\text{min}} = 2 \), \( y_{\text{max}} = 16 \); b) \( y_{\text{min}} = 1 \), \( y_{\text{max}} = 2 \).