To determine the leading coefficient of a polynomial function, we need to identify the term with the highest exponent of [tex]\( x \)[/tex]. The leading coefficient is the coefficient of this highest-degree term.
Let's analyze the polynomial function:
[tex]\[
F(x) = \frac{1}{4} x^5 + 8 x - 5 x^4 - 19
\][/tex]
1. Identify the term with the highest power of [tex]\( x \)[/tex]:
- The terms in the polynomial are:
- [tex]\(\frac{1}{4} x^5\)[/tex]
- [tex]\(8 x\)[/tex]
- [tex]\(-5 x^4\)[/tex]
- [tex]\(-19\)[/tex]
- Among these terms, [tex]\(\frac{1}{4} x^5\)[/tex] has the highest power of [tex]\( x \)[/tex], which is [tex]\( 5 \)[/tex].
2. Determine the coefficient of this leading term:
- The leading term [tex]\(\frac{1}{4} x^5\)[/tex] has a coefficient of [tex]\(\frac{1}{4}\)[/tex].
3. Conclusion:
- The coefficient of the term with the highest power of [tex]\( x \)[/tex] (which is [tex]\( x^5 \)[/tex]) is the leading coefficient.
Therefore, the leading coefficient of the polynomial [tex]\( F(x) = \frac{1}{4} x^5 + 8 x - 5 x^4 - 19 \)[/tex] is:
[tex]\[
\boxed{\frac{1}{4}}
\][/tex]
So, the correct answer is option [tex]\( D \)[/tex].
