For the function 𝑔(𝑡)=4𝑡4−4𝑡g(t)=4t 4 −4 t , which of the following statements are true?I. lim𝑡→0𝑔(𝑡)=−1II. lim𝑡→∞𝑔(𝑡)=−∞III. 𝑔(𝑡) has 2 roots. I. t→0lim g(t)=−1II. t→∞lim g(t)=−∞III. g(t) has 2 roots. A.I onlyB.II onlyC.III onlyD.I and II onlyE.I, II, and III
Question
For the function 𝑔(𝑡)=4𝑡4−4𝑡g(t)=4t 4 −4 t , which of the following statements are true?I. lim𝑡→0𝑔(𝑡)=−1II. lim𝑡→∞𝑔(𝑡)=−∞III. 𝑔(𝑡) has 2 roots. I. t→0lim g(t)=−1II. t→∞lim g(t)=−∞III. g(t) has 2 roots. A.I onlyB.II onlyC.III onlyD.I and II onlyE.I, II, and III
Solution
The function given is g(t) = 4t^4 - 4t. Let's evaluate each statement one by one.
I. lim t→0 g(t) = -1 When we substitute t=0 into the function, we get g(0) = 4*(0)^4 - 4*(0) = 0. So, the first statement is false.
II. lim t→∞ g(t) = -∞ As t approaches infinity, the term 4t^4 will dominate over -4t. Since 4t^4 goes to positive infinity as t goes to infinity, the limit of the function as t approaches infinity is positive infinity, not negative infinity. So, the second statement is also false.
III. g(t) has 2 roots The roots of the function are the values of t that make g(t) = 0. Setting the function equal to zero gives us 4t^4 - 4t = 0. We can factor this to get 4t(t - 1)(t + 1)(t^2 + 1) = 0. This gives us three real roots: t = 0, t = 1, and t = -1. The term t^2 + 1 = 0 gives us two complex roots, but these are not real roots. So, the third statement is true.
Therefore, the correct answer is C. III only.
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