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Application of Derivatives

Multiple Choice Questions

211.

The function f(x) = $\log (1 + x) - \frac{2 x}{2 + x}$ is increasing on

$(- 1, \infty)$
$(- \infty, 0)$
$(- \infty, \infty)$
None of these

212.

The minimum value of $x^{2} + \frac{1}{1 + x^{2}}$ is at

x = 0
x = 1
x = 4
x = 3

213.

The maximum value of $3 \cos (θ) + 4 \sin (θ)$ is

3
4
5
None of these

214.
The function x⁵ - 5x⁴ + 5x³ - 1 is

neither maximum nor minimum at x = 0

maximum at x = 0

maximum at x = 1 and minimum at x = 3

minimum at x = 0

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maximum at x = 1 and minimum at x = 3

$Let f (x) = x^{5} - 5 x^{4} + 5 x^{3} - 1 On differentiating w . r . t . x, we get f' (x) = 5 x^{4} - 20 x^{3} + 15 x^{2} For maximum or minimum, put f' (x) = 0 5 x^{4} - 20 x^{3} + 15 x^{2} = 0 \Rightarrow x^{2} (5 x^{2} - 20 x + 15) = 0 \Rightarrow 5 x^{2} (x - 1) (x - 3) = 0 \Rightarrow x = 0, 1, 3 Again differentiating w . r . t . x, we get f'' (x) = 5 (4 x^{3} - 12 x^{2} + 6 x) At x = 1, f'' (1) = 5 (4 - 12 + 6) = - 10 < 0, maximum At x = 3, f'' (3) = 5 (4 \times 27 - 12 \times 9 + 6 \times 3) = 90 > 0, minimum At x = 0, f'' (0) = 5 (0 - 12 \times 0 + 6 \times 0) = 0, we have further check f''' (x) = 5 (12 x^{2} - 24 x + 6) \Rightarrow f''' (0) = 30 \neq 0 [Inflexion] ∴ f (x) is maximum at x = 1 and minimum at x = 3 .$

215.

If the radius of a circle be increasing at a uniform rate of 2 cm/s. The area of increasing of area of circle, at the instant when the radius is 20 cm, is