(e) Every functionf defined on (-∞, ∞) that satisfies the condition that lim f(x) = lim f(x) = ∞ must have at least x00 X -00 one critical point. True False
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- If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,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.5. Consider the function f(x) = |x| + 2 at x = 0. Provide a table of inputs and outputs that demonstrate the limitdefinition of the derivative, numerically
- 2. Give an example showing that, even if f0(c) = 0, it could happen that f(x) does not have a localmaximum or minimum at x = c. How is this consistent with Fermat’s Theorem in the book?The limit of (xy-2y) / (x2+y2-4x+4) as (x,y) approaches (2,0) is solved and found out that it does not exist. Can we make this continuous by defining f(2,0) = k for some real value k? If yes, what could this value be? If no, why not?For the continuous function f(x), if f’(-4) does not exist, then x=-4 is a critical point.
- We consider the rational function f(x) = p(x)/ q(x) = 2x ^2 + cx − 4c / x^2 − x − 2 . Here c is an unknown constant. If the limit L = limx→2 f(x) is a finite number, then what is c?If the above graph is the graph of f′(x), then f(x) has a local maximum at x=___, a local minimum at x=____ and a critical point that is neither a local maximum nor a local minimum at x=____Find g'(x) for g(x)=-x^2+3x+2 by using the limit definition of a derivative. lim x-->0 f(x+h)-f(x)/h