1) True statement: by first derivative test f'(a) = 0 implies 'a' is a loca extremum point.
2) True statement: consider the graph of the function. Draw a parallel line to X-axis. Move the line up and down to get a tangent point to the line. This tangent point will be an extremum.
3) False statement: This need not be true always. For example let f(x) = 1-x in [0,1] , then x = 0 is a local maximum. Now f²(x) = f(f(x)) = f(1-x) = 1-(1-x) = x. But x = 0 is not a local maximum for f²(x) in [0,1].
4) True statement: since 'a' is a local minimum of both f and g we have f'(a) = 0 = g'(a). Now (f+g)'(a) = f'(a)+g'(a) = 0. Thus 'a' is an extremum point of f+g and it will be a local minimum as 'a' is individually local minimum.
5) True statement: since f is decreasing on (a,b) the slope of the curve is negative. Slope of the function curve at x is f'(x) hence f'(x) < 0 on (a,b).
Put (v) for true and (x) for false sentences 1. () If f'(a) 0, then f (a) is a local extrema of f...
4. True or False. Write true or false in the blanks. a, A continuous function over a closed interval will achieve exactly one local maximum on that interval ______________ b. If f(x) and g(x) both have a local maximum at x=a then has either a local maximum or a local minimum at x=a. ___________ c. If for all x and if a > b, then _____________ d. If is undefined, and if is continuous at x=c, then has a local...
1. Determine whether the statement is true or false. If false, explain why and correct the statement (T/FIf)exists, then lim ()f) o( T / F ) If f is continuous, then lim f(x) = f(r) (TFo)-L, then lim f(x)- lim F(x) "( T / F ) If lim -f(x)s lim. f(x) L, then lim f(x)s 1. "(T/F) lim. In x -oo . (T/F) lim0 ·(T / F ) The derivative f' (a) is the instantaneous rate of change of y...
Consider the following graph of f(x) on the closed interval (0,5): 5 4 3 2 1 0 -1 0 1 2 3 5 6 (If the picture doesn't load, click here 95graph2) Use the graph of f(x) to answer the following: (a) On what interval(s) is f(x) increasing? (b) On what interval(s) is f(x) decreasing? (c) On what interval(s) is f(x) concave up? (d) On what interval(s) is f(x) concave down? (e) Where are the inflection points (both x and...
Use implicit differentiation to find y' and evaluate y'at (-1,-6). 9xy + y-48 = 0 y' = Evaluate y' at (-1,-6). y'l(-1,-6)=0 (Simplify your answer.) Find the intervals on which f(x) is increasing, the intervals on which f(x) is decreasing, and the local extrema. f(x) = x + 10x + 21 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The function is increasing on (Type your answer using interval...
true or false The real valued function f : (1,7) + R defined by f(x) = 2is uniformly contin- uous on (0,7). Let an = 1 -1/n for all n € N. Then for all e > 0) and any N E N we have that Jan - am) < e for all n, m > N. Let f :(a,b) → R be a differentiable function, if f'() = 0 for some point Xo € (a, b) then X, is...
Consider the function on the interval (0, 2). f(x) = sin(x) cos(x) + 8 (a) Find the open interval(s) on which the function is increasing or decreasing. (Enter your answers using interval notation.) increasing decreasing (b) Apply the First Derivative Test to identify all relative extrema. relative maxima (x, y) = (smaller x-value) (larger x-value) (X,Y)= (x, y) = (1 (x, y) = relative minima (smaller x-value) (larger x-value)
Please let me know whether true or false If false, please give me the counter example! (a) If a seriesE1an converges, then lim,n-0 an = 0. m=1 (b) If f O(g), then f(x) < g(x) for all sufficiently large . R is any one-to-one differentiable function, then f-1 is (c) If f: R differentiable on R (d) The sequence a1, a2, a3, -.. defined by max{ sin 1, sin 2,-.- , sin n} an converges (e) If a power series...
Determine whether the statement is TRUE or FALSE. You are NOT required to justify your answers. (a) Suppose both f and g are continuous on (a, b) with f > 9. If Sf()dx = Sº g(x)dx, then f(x) = g(x) for all 3 € [a, b]. (b) If f is an infinitely differentiable function on R with f(n)(0) = 0 for all n = 0,1,2,..., then f(x) = 0 for all I ER. (c) f is improperly integrable on (a,...
True or False: If f(x) and g(x) are two differentiable functions on an interval (a,b), and f(x)>g(x) on (a,b), then f'(x)>g'(x).
True of False (dont need to solve) _____According to the L’hospital’s rule, lim(x-->a) f(x)/g(x)= L if f'(x)/g'(x)=L _____ A monotonically increasing function cannot have a max or min in a closed interval [a, b] because f’(x) >0 for all x in (a, b). _____ For x>10, the ratio ln(x)/sqrt(x-10) approaches 0 as x --> infinity . _____ lim(x-->infinity) (3x^3 +12x^2 -8x +1)/(x^2 + 2)(3x-1) =3 _____ _____ x = -1 is an asymptote of y= 1/(sqrt(x+1)) _____ _____ y=0 is...