To find the value of [tex]\( f(x) \)[/tex] at [tex]\( x = \frac{1}{2} \)[/tex] given the function [tex]\( f(x) = 2x^3 - x^2 + 32x - 16 \)[/tex], follow these steps:
1. Substitute [tex]\( x = \frac{1}{2} \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f\left(\frac{1}{2}\right) = 2\left(\frac{1}{2}\right)^3 - \left(\frac{1}{2}\right)^2 + 32\left(\frac{1}{2}\right) - 16
\][/tex]
2. Calculate each term separately:
- For the term [tex]\( 2\left(\frac{1}{2}\right)^3 \)[/tex]:
[tex]\[
\left(\frac{1}{2}\right)^3 = \frac{1}{8}
\][/tex]
[tex]\[
2 \cdot \frac{1}{8} = \frac{2}{8} = \frac{1}{4}
\][/tex]
- For the term [tex]\( \left(\frac{1}{2}\right)^2 \)[/tex]:
[tex]\[
\left(\frac{1}{2}\right)^2 = \frac{1}{4}
\][/tex]
- For the term [tex]\( 32\left(\frac{1}{2}\right) \)[/tex]:
[tex]\[
32 \cdot \frac{1}{2} = 16
\][/tex]
- The constant term is [tex]\( -16 \)[/tex]:
3. Combine all these results:
[tex]\[
f\left(\frac{1}{2}\right) = \frac{1}{4} - \frac{1}{4} + 16 - 16
\][/tex]
4. Simplify the expression:
[tex]\[
f\left(\frac{1}{2}\right) = \frac{1}{4} - \frac{1}{4} + 16 - 16 = 0
\][/tex]
Therefore, the value of the function [tex]\( f(x) \)[/tex] at [tex]\( x = \frac{1}{2} \)[/tex] is [tex]\( 0 \)[/tex].
