we have the function
[tex]f(x)=7x^5-9x^4-x^2[/tex]Simplify the expression
[tex]f(x)=x^2(7x^3-9x^2-1)[/tex]Find out the value of f(x) at x=1
For x=1
[tex]\begin{gathered} f(x)=7(1)^5-9(1)^4-(1)^2 \\ f(x)=7-9-1 \\ f(x)=-3 \end{gathered}[/tex]Find out the value of f(x) at x=2
For x=2
[tex]\begin{gathered} f(x)=7(2)^5-9(2)^4-(2)^2 \\ f(x)=224-144-4 \\ f(x)=76 \end{gathered}[/tex]Note that
For x=1 --------> f(x) is negative
For x=2 ------> f(x) is positive
that means
between the interval (1,2) the graph cross the x-axis
that means
The given function has at least one real zeros between x=1 and x=2