Question

Consider the function f:R→R defined by f(x) = (x² – ax + 1)/(x² + ax + 1), where 0 < a < 2

(i) which of the following is true?

(a) (2 + a)² f''(1) + (2 – a)² f''(–1) = 0

(b) (2 – a)² f''(1) – (2 – a)² f''(–1) = 0

(c) f'(1)f'(–1) = (2 – a)²

(d) f'(1)f'(–1) = (2 + a)²

(ii) which of the following is true?

(a) f(x) is a decreasing on (–1, 1) and has a local minimum at x = 1.

(b) f(x) is a increasing on (–1, 1) and has a local maximum at x = 1.

(c) f(x) is increasing on (–1, 1) and has neither local maximum nor a local minimum at x = 1

(d) f(x) is decreasing on (–1, 1) and has neither local maximu nor a local minimum at x = 1.

(iii) Let g(x) = ∫(f'(t)/(1 + t²))dt for t ∈ [0, e^x].

Which of the following is true?

(a) g'(x) is +ve on (–∞, 0) and –ve on (0, ∞)

(b) g'(x) is –ve on (–∞, 0) and +ve on (0, ∞)

(c) g'(x) changes sign on both (–∞, 0) and (0, ∞)

(d) g'(x) does not change sign on (–∞, ∞)

Answer 1

Correct option  (i) (a), (ii) (a), (iii) (b)

Explanation:

(i)

From the sign scheme of f'(x), we get, f(x) decreasing in (–1, 1) and a local minima at x = 1.

(iii)

From the sign scheme, it is clear that, g'(x) is positive in (0, ∞) and negative in (–∞, 0)