Two of the following statements are true. Select the true statements. 5. If \\(f
ID: 2905540 • Letter: T
Question
Two of the following statements are true. Select the true statements.
5.
If (f(x) = (5-x) sqrt[3]{x-1}), find ( f all ) critical numbers of (f(x).)
6.
For the function ({displaystyle f(x) = rac{x}{x^2+1} }), find the absolute maximum value (M) and the absolute minimum value (m) on the interval ({ displaystyle left[ 0, 3 ight]. })
7.
Verify that the function (f(x) = sqrt[3]{x} ) satisfies the hypotheses of the Mean Value Theorem on the interval ( [0,8] ) .
Then find all numbers (c) that satisfy the conclusion of the Mean Value Theorem.
8.
For the given function ( {displaystyle f(x) = x^{ rac{2}{3}} (x+4) }), which of the following statement is true?
Two of the following statements are true. Select the true statements.
5.
If (f(x) = (5-x) sqrt[3]{x-1}), find ( f all ) critical numbers of (f(x).)
6.
For the function ({displaystyle f(x) = rac{x}{x^2+1} }), find the absolute maximum value (M) and the absolute minimum value (m) on the interval ({ displaystyle left[ 0, 3 ight]. })
7.
Verify that the function (f(x) = sqrt[3]{x} ) satisfies the hypotheses of the Mean Value Theorem on the interval ( [0,8] ) .
Then find all numbers (c) that satisfy the conclusion of the Mean Value Theorem.
8.
For the given function ( {displaystyle f(x) = x^{ rac{2}{3}} (x+4) }), which of the following statement is true?
Two of the following statements are true. Select the true statements.
B) If (f'(x) < 0) for (x in (3, 6)), then (f(x)) is decreasing ((3, 6)).
C) If (f''(x) > 0) for (x in (3, 6)), then (y = f(x)) has a local minimum on ((3, 6)).
D) If (f''(x) > 0) for (x in (3, 6)), then (y = f(x)) is concave up on ((3, 6)).
E) If (f''(x) < 0) for (x in (3, 6)), then (y = f(x)) has an inflection point on ((3, 6)).
5.
If (f(x) = (5-x) sqrt[3]{x-1}), find ( f all ) critical numbers of (f(x).)
B) (x=-1,1 ) only
C) (x=3, 5) only
D) (x=1, 5 ) only
E) (x=1, 2 ) only
6.
For the function ({displaystyle f(x) = rac{x}{x^2+1} }), find the absolute maximum value (M) and the absolute minimum value (m) on the interval ({ displaystyle left[ 0, 3 ight]. })
B) ({ displaystyle m= - rac{1}{2}, M= rac{1}{10} })
C) ({ displaystyle m= 0, M= rac{1}{2} })
D) ({ displaystyle m= - rac{1}{2}, M= rac{3}{10} })
E) ({ displaystyle m= - rac{1}{2}, M= rac{1}{2} })
7.
Verify that the function (f(x) = sqrt[3]{x} ) satisfies the hypotheses of the Mean Value Theorem on the interval ( [0,8] ) .
Then find all numbers (c) that satisfy the conclusion of the Mean Value Theorem.
B) ({ displaystyle c= rac{8sqrt{3}}{9} })
C) ({ displaystyle c= rac{64}{27} })
D) ({ displaystyle c= pm rac{8sqrt{3}}{9} })
E) (f(x)) does not satisfy the hypotheses
8.
For the given function ( {displaystyle f(x) = x^{ rac{2}{3}} (x+4) }), which of the following statement is true?
B) (f) is increasing on ( {displaystyle left( - rac{8}{5}, 0 ight) })
C) (f) is concave up on ( {displaystyle left(- infty, rac{4}{5} ight) })
D) (f) is concave down on ( {displaystyle left(- infty, 0 ight) cup left(0, rac{4}{5} ight) })
E) (f) is has an inflection point at ( x=0 ) and ( x = dfrac{4}{5} )
Explanation / Answer
1. Option C and D are True
Point for local minima is obtained if a function at a point xo gives f'(x) =0 and f"(x)>0
If f ''(x) exists at x0 and is positive, then f ''(x)is concave up at x0. If f ''(x0) exists and is negative, then f(x) is concue ave down at x0.
6.
f(x) = x/(x^2 +1)
f'(x) = 2x/(x^2+1)^2
f'(x) =0 ; x=0
Now we check f(x) at x=0 , 0, 3
f0) =0; f(3) =3/10
m=0 and M= 3/10
Option A
7. By mean value theorem f'(c) = f(b)-f(a)/(b-a) if f(x) is continous in the interval (a,b) and differentiable in the interval(a,b)
f(x) = sqrt(3x)
f'(c) = 1/2 sqrt3/sqrtc and {f(b)-f(a)}/(b-a) = sqrt(24)/8
1/2 sqrt3/sqrtc = sqrt(24)/8
solving for c we get c=0,2
option A
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