82. The derivative of the function ƒ is given by f'(x) = e¯*cos(x2), for all real numbers x. What is the minimum value of f(x) for - 1 ≤ x ≤1? (A) f(-1) (B) f(-0.762) (C) f(1) (D) There is no minimum value of f(x) for -1 ≤ x ≤ 1.
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- Does a Limiting Value Occur? A rocket ship is flying away from Earth at a constant velocity, and it continues on its course indefinitely. Let D(t) denote its distance from Earth after t years of travel. Do you expect that D has a limiting value?On the graph of f(x)=sinx and the interval [−2π,0), for what value of x does f(x) achieve a minimum?Shown is the graph of the derivative f'(x) of the function f(x). (a) Identify the critical numbers of f(x). (b) For each critical value, determine whether it is a local maximum, local minimum, or neither. (c) Is f''(−2) positive or negative? What can you say about the concavity of the original function f(x) nearby x = −2?
- Suppose g(x) is a differentiable function such that g′(3) = 0 and g′′(3) = 0. Then the Second Derivative Test guarantees that x = 3 is neither a local maximum nor a local minimum. Explain whyGive an example of a differentiable function ƒ whose first derivative is zero at some point c even though ƒ has neither a local maximum nor a local minimum at c.2. Suppose that f(x) is a function continuous for every value of x whose first derivative is f'(x) = 2(1-x)/1+x^2 and f"(x)= 4x(x^2-3)/ (1+x^2)^2 Further, assume that it is known that f has a horizontal asymptote at y = 0. and a. Determine all critical points of f.
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