\( 1<\quad \) Use the Intermediate Value Theorem to show that the polynomial \( f(x)=4 x^{4}-7 x^{2}+1 \) has a real zero between -1 and 0 , Select the correct choice below and fill in the answer boxes to complete your choice. A. Because \( f(x) \) is a polynomial with \( f(-1)=\square>0 \) and \( f(0)=\square>0 \), the function has a real zero between -1 and 0 . 0 of 8 O B. Because \( f(x) \) is a polynomial with \( f(-1)=\square<0 \) and \( f(0)=\square<0 \), the function has a real zero between -1 and 0 . O C. Because \( f(x) \) is a polynomial with \( f(-1)=\square>0 \) and \( f(0)=\square<0 \), the function has a real zero between \( =1 \) and 0 . O D. Because \( f(x) \) is a polynomial with \( f(-1)=\square<0 \) and \( f(0)=\square>0 \), the function has a real zero between -1 and 0 .

To apply the Intermediate Value Theorem, let's first evaluate the polynomial at \( x = -1 \) and \( x = 0 \): 1. Calculate \( f(-1) \): \[ f(-1) = 4(-1)^4 - 7(-1)^2 + 1 = 4(1) - 7(1) + 1 = 4 - 7 + 1 = -2 \] 2. Calculate \( f(0) \): \[ f(0) = 4(0)^4 - 7(0)^2 + 1 = 1 \] Now we can fill in the answer boxes. The correct choice is: D. Because \( f(x) \) is a polynomial with \( f(-1)=-2<0 \) and \( f(0)=1>0 \), the function has a real zero between -1 and 0. --- A little historical tidbit: The Intermediate Value Theorem has roots that trace back to 17th-century mathematicians like Descartes and Rolle. They were pioneers in the study of calculus and laid the groundwork for understanding continuous functions. For real-world application, think about how the Intermediate Value Theorem is used in computer science, particularly in algorithms for root-finding. When searching for solutions in problems like optimizing a function, this theorem helps programmers ensure they can find solutions within defined intervals, making it a key tool in numerical methods!