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Removable and Nonremovable Discontinuities In Exercises 41-60, find the x- values (if any) at which f is not continuous. Which of the discontinuities are removable?
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Calculus: Early Transcendental Functions
- Rolle's theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b. Is it True or False?arrow_forwardh5. Prove that f ( x ) = x ⋅ | x | is continuous at all points c in ℝ.arrow_forwardAdvanced Calculus: Use the Bolzano–Weierstrass Theorem to prove that if f is a continuous function on [a,b], then f is bounded on [a,b] (that is, there exists M > 0 such that |f(x)| ≤ M for all x ∈[a,b]). (Hint: Give a proof by contradiction.)arrow_forward
- Determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply.) f(x) = (x − 3)(x − 4)(x − 8), [3, 8] Yes, Rolle's Theorem can be applied. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in the open interval (a, b). No, because f(a) ≠ f(b). If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f '(c) = 0. (Enter your answers as a comma-separated list. If Rolle's Theorem cannot be applied, enter NA.) c=arrow_forwardDetermining Concavity In Exercises 3–14,determine the open intervals on which the graphof the function is concave upward or concavedownward.' 3. f (x) = x2 − 4x + 8arrow_forwardProve the Cauchy Mean Value Theorem. Let f and h be real-valued functions continuous on [a, b], differentiable on (a, b), and h(a) not equal to h(b). There exists c in (a, b) such that (f(b) - f(a))h'(c) = f'(c)(h(b)-h(a)).arrow_forward
- Determine if the statement is true or false. Provide a counterexample or proof. Suppose f is a continuous injective function, and f-1 is differentiable, then f is also differentiable.arrow_forwardDetermine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply.) f(x) = −x2 + 7x, [0, 7] Yes, Rolle's Theorem can be applied. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in the open interval (a, b). No, because f(a) ≠ f(b). If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f '(c) = 0. (Enter your answers as a comma-separated list. If Rolle's Theorem cannot be applied, enter NA.) c =arrow_forwardLet f be a continuous function on [a,b]⊂R and f(x)∈[a,b] for all x∈[a,b]. Prove that there exists a point c in [a,b] at which f(c) =c. We call point c a fixed point of f. Hint: Apply the intermediate-value theorem to the function g(x) =f(x)−xarrow_forward
- Applying the continuity theorems For what values of x is the function ƒ1x2= x / x2 - 7x + 12 continuous?arrow_forwardRemovable and Nonremovable Discontinuities. Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable?arrow_forwardDetermine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The Mean Value Theorem can be applied to f(x) = 1/x on the interval [−1, 1].arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage