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Consider two continuous functions f and g defined on the interval [-1, 2], which are also twice differentiable on (-1, 2). The values of f and g at x = -1, 0, and 2 are provided in the table below:
x = -1 x = 0 x = 2
f(x) = 3 f(x) = 6 f(x) = 0
g(x) = 0 g(x) = 1 g(x) = -1
In the intervals (-1, 0) and (0, 2), the second derivative of (f - 3g), denoted as (f - 3g)'', does not become zero at any point. Which of the following statements is true?
- The equation f'(x) - 3g'(x) = 0 has exactly three solutions in the union of (-1, 0) and (0, 2).
- The equation f'(x) - 3g'(x) = 0 has exactly one solution in the interval (-1, 0).
- The equation f'(x) - 3g'(x) = 0 has exactly one solution in the interval (0, 2).
- The equation f'(x) - 3g'(x) = 0 has exactly two solutions in (-1, 0) and exactly two solutions in (0, 2).
Correct answer: The equation f'(x) - 3g'(x) = 0 has exactly one solution in the interval (-1, 0).
Solution
The second derivative of (f - 3g) does not become zero, indicating a single turning point in each interval. By Rolle's theorem, f'(x) - 3g'(x) = 0 has exactly one solution in (-1, 0).
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- Column 1 contains details about the zeros of f(x), f'(x), and f''(x).
- Column 2 contains information about the behavior of f(x), f'(x), and f''(x) as x approaches infinity.
- Column 3 contains details about the increasing or decreasing nature of f(x) and f'(x).
Column 1:
(I) f(x) = 0 for some x in (1, e²)
(II) f'(x) = 0 for some x in (1, e)
(III) f'(x) = 0 for some x in (0, 1)
(IV) f''(x) = 0 for some x in (1, e)
Column 2:
(i) lim x→∞ f(x) = 0
(ii) lim x→∞ f(x) = −∞
(iii) lim x→∞ f'(x) = −∞
(iv) lim x→∞ f''(x) = 0
Column 3:
(P) f is increasing on (0, 1)
(Q) f is decreasing on (e, e²)
(R) f' is increasing on (0, 1)
(S) f' is decreasing on (e, e²)
Which of the following options gives the correct combination?
- Let f: R → R be a function that is twice differentiable, satisfying f''(x) > 0 for every x in R, with f(1/2) = 1/2 and f(1) = 1. Which of the following is true about f'(1)?
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