Answer:
Smaller t = 2
Larger t = 5
Step-by-step explanation:
Given:
The given function is.
[tex]f(t)=-(t-2)(t-5)[/tex]
Find the zeros of the function.
Solution:
[tex]f(t)=-(t-2)(t-5)[/tex]
Simplify the equation above equation.
[tex]f(t)=-(t^{2}-5t-2t+10)[/tex]
[tex]f(t)=-(t^{2}-7t+10)[/tex]
[tex]f(t)=-t^{2}+7t-10[/tex]
Now, we first find the root of the above equation.
Use quadratic formula with [tex]a=-1, b=7, c=-10[/tex].
[tex]t=\frac{-b\pm \sqrt{(b)^{2}-4ac}}{2a}[/tex]
Put a, b and c value in above equation.
[tex]t=\frac{-7\pm \sqrt{(7)^{2}-4(-1)(-10)}}{2(-1)}[/tex]
[tex]t=\frac{-7\pm \sqrt{49-4\times 10}}{-2}[/tex]
[tex]t=\frac{-7\pm \sqrt{49-40}}{-2}[/tex]
[tex]t=\frac{-7\pm \sqrt{9}}{-2}[/tex]
[tex]t=\frac{-7\pm 3}{-2}[/tex]
For positive sign
[tex]t=\frac{-7 + 3}{-2}[/tex]
[tex]t=\frac{-4}{-2}[/tex]
t = 2
For negative sign
[tex]t=\frac{-7 - 3}{-2}[/tex]
[tex]t=\frac{-10}{-2}[/tex]
t = 5
Put t = 2 in given function.
[tex]f(t)=-(2-2)(2-5)=0[/tex]
Put t = 5 in given function.
[tex]f(t)=-(5-2)(5-5)=0[/tex]
So, the zeros of the function is t = 2 or 5
Therefore, the smaller value of t = 2 and larger value of t = 5.
