Advertisements
Advertisements
Question
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Options
a ∈ (1/2, ∞)
a ∈ (−1/2, 1/2)
a = 1/2
a ∈ R
Advertisements
Solution
\[f(x) = 2 \tan x + \left( 2a + 1 \right) \log_e \left| \sec x \right| + \left( a - 2 \right) x\]
\[\text { When }\sec x > 0 \Rightarrow \left| \sec x \right| = \sec x\]
\[\frac{d}{dx}\left\{ f\left( x \right) \right\} = 2 \sec^2 x + \left( 2a + 1 \right)\frac{1}{\sec x} \times \sec x \tan x + \left( a - 2 \right) \]
\[ = 2 \sec^2 x + \left( 2a + 1 \right)\tan x + \left( a - 2 \right) \]
\[\text { For f(x) to be increasing}, \]
\[2se c^2 x + \left( 2a + 1 \right)\tan x + \left( a - 2 \right) \geqslant 0\]
\[ \Rightarrow 2 + 2 \tan^2 x + \left( 2a + 1 \right)\tan x + a - 2 \geqslant 0\]
\[ \Rightarrow 2 \tan^2 x + \left( 2a + 1 \right)\tan x + a \geqslant 0\]
\[\text { Its discriminant } \leqslant 0 \left[ \because a x^2 + bx + c \geqslant 0 \Rightarrow b^2 - 4ac \leqslant 0 \right]\]
\[ \Rightarrow \left( 2a + 1 \right)^2 - 4 . 2 . a \leqslant 0\]
\[ \Rightarrow 4 a^2 - 4a + 1 \leqslant 0\]
\[ \Rightarrow \left( 2a - 1 \right)^2 \leqslant 0\]
\[ \left( 2a - 1 \right)^2 < 0 \text { cannot be possible } . \]
\[ \therefore \left( 2a - 1 \right)^2 = 0\]
\[ \Rightarrow a = \frac{1}{2}\]
\[\text { When } \sec x < 0 \Rightarrow \left| \sec x \right| = - \sec x\]
\[\frac{d}{dx}\left\{ f\left( x \right) \right\} = 2 \sec^2 x + \left( 2a + 1 \right)\frac{1}{- \sec x} \times \sec x \tan x + \left( a - 2 \right)\]
\[ = 2 \sec^2 x - \left( 2a + 1 \right)\tan x + \left( a - 2 \right) \]
\[\text { For f(x) to be increasing,} \]
\[2se c^2 x - \left( 2a + 1 \right)\tan x + \left( a - 2 \right) \geqslant 0\]
\[ \Rightarrow 2 + 2 \tan^2 x - \left( 2a + 1 \right)\tan x + a - 2 \geqslant 0\]
\[ \Rightarrow 2 \tan^2 x - \left( 2a + 1 \right)\tan x + a \geqslant 0 \]
\[\text { Its discriminant } \leqslant 0 \left[ \because a x^2 + bx + c \geqslant 0 \Rightarrow b^2 - 4ac \leqslant 0 \right]\]
\[ \Rightarrow \left\{ - \left( 2a + 1 \right) \right\}^2 - 4 . 2 . a \leqslant 0\]
\[ \Rightarrow 4 a^2 - 4a + 1 \leqslant 0\]
\[ \Rightarrow \left( 2a - 1 \right)^2 \leqslant 0\]
\[ \left( 2a - 1 \right)^2 < 0 \text { cannot be possible } . \]
\[ \therefore \left( 2a - 1 \right)^2 = 0\]
\[ \Rightarrow a = \frac{1}{2}\]
APPEARS IN
RELATED QUESTIONS
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Find the interval in which the following function are increasing or decreasing f(x) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 − 9x − x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
The function f(x) = cot−1 x + x increases in the interval
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
Find `dy/dx,if e^x+e^y=e^(x-y)`
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
y = log x satisfies for x > 1, the inequality ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
As one moves from left to right on a graph, what does a decrease in the \[y\]-values indicate?
