Question
Let f be a function such that f (1) = − 2 and f (5) = 7. Which of the following conditions ensures that f(c) = 0 for some value c in the open interval (1,5)?
A \(\int ^{5}_{1}f(x)dx\) exists.
B f is increasing on the closed interval [1, 5].
C f is continuous on the closed interval [1, 5].
D f is defined for all values of x in the closed interval [1, 5].
Answer/Explanation
Question
Let f be the function given by \(f(x)=2x+\tan (\frac{x}{3})-15 \). The Intermediate Value Theorem applied to f on the closed interval [10,15] guarantees a solution in [10,15] to which of the following equations?
Af(x)=−15
B f(x)=0
C f(x)=5
D f(x)=15
Answer/Explanation
Ans:C
Question
Let f be a function such that f(3)<4<f(5). Which of the following statements provides sufficient additional information to conclude that there is a value x=c in the interval [3,5] such that f(c)=4 ?
A f is defined for all x.
B f is increasing for all x.
C f is continuous for all x.
D There is a value x=c in the interval [3,5] such that \(\lim_{x\rightarrow c}f(x)=4\).
Answer/Explanation
Ans:C
Question
Let f be a function of x. Which of the following statements, if true, would guarantee that there is a number c in the interval [−5,4] such that f(c)=12?
A f is increasing on the interval [−5,4] , where f(−5)=0 and f(4)=20 .
B f is increasing on the interval [−5,4], where f(−5)=15 and f(4)=30.
C f is continuous on the interval [−5,4], where f(−5)=0 and f(4)=20.
D f is continuous on the interval [−5,4], where f(−5)=15 and f(4)=30.
Answer/Explanation
Ans:C