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 value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
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 \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Show that f(x) = e2x is increasing on R.
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
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 'a' for which f(x) = loga x is increasing in its domain ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
y = x(x – 3)2 decreases for the values of x given by : ______.
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
