Advertisements
Advertisements
प्रश्न
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
पर्याय
a ∈ (1/2, ∞)
a ∈ (−1/2, 1/2)
a = 1/2
a ∈ R
Advertisements
उत्तर
\[f(x) = 2 \tan x + \left( 2a + 1 \right) \log_e \left| \sec x \right| + \left( a - 2 \right) x\]
\[\text { When }\sec x > 0 \Rightarrow \left| \sec x \right| = \sec x\]
\[\frac{d}{dx}\left\{ f\left( x \right) \right\} = 2 \sec^2 x + \left( 2a + 1 \right)\frac{1}{\sec x} \times \sec x \tan x + \left( a - 2 \right) \]
\[ = 2 \sec^2 x + \left( 2a + 1 \right)\tan x + \left( a - 2 \right) \]
\[\text { For f(x) to be increasing}, \]
\[2se c^2 x + \left( 2a + 1 \right)\tan x + \left( a - 2 \right) \geqslant 0\]
\[ \Rightarrow 2 + 2 \tan^2 x + \left( 2a + 1 \right)\tan x + a - 2 \geqslant 0\]
\[ \Rightarrow 2 \tan^2 x + \left( 2a + 1 \right)\tan x + a \geqslant 0\]
\[\text { Its discriminant } \leqslant 0 \left[ \because a x^2 + bx + c \geqslant 0 \Rightarrow b^2 - 4ac \leqslant 0 \right]\]
\[ \Rightarrow \left( 2a + 1 \right)^2 - 4 . 2 . a \leqslant 0\]
\[ \Rightarrow 4 a^2 - 4a + 1 \leqslant 0\]
\[ \Rightarrow \left( 2a - 1 \right)^2 \leqslant 0\]
\[ \left( 2a - 1 \right)^2 < 0 \text { cannot be possible } . \]
\[ \therefore \left( 2a - 1 \right)^2 = 0\]
\[ \Rightarrow a = \frac{1}{2}\]
\[\text { When } \sec x < 0 \Rightarrow \left| \sec x \right| = - \sec x\]
\[\frac{d}{dx}\left\{ f\left( x \right) \right\} = 2 \sec^2 x + \left( 2a + 1 \right)\frac{1}{- \sec x} \times \sec x \tan x + \left( a - 2 \right)\]
\[ = 2 \sec^2 x - \left( 2a + 1 \right)\tan x + \left( a - 2 \right) \]
\[\text { For f(x) to be increasing,} \]
\[2se c^2 x - \left( 2a + 1 \right)\tan x + \left( a - 2 \right) \geqslant 0\]
\[ \Rightarrow 2 + 2 \tan^2 x - \left( 2a + 1 \right)\tan x + a - 2 \geqslant 0\]
\[ \Rightarrow 2 \tan^2 x - \left( 2a + 1 \right)\tan x + a \geqslant 0 \]
\[\text { Its discriminant } \leqslant 0 \left[ \because a x^2 + bx + c \geqslant 0 \Rightarrow b^2 - 4ac \leqslant 0 \right]\]
\[ \Rightarrow \left\{ - \left( 2a + 1 \right) \right\}^2 - 4 . 2 . a \leqslant 0\]
\[ \Rightarrow 4 a^2 - 4a + 1 \leqslant 0\]
\[ \Rightarrow \left( 2a - 1 \right)^2 \leqslant 0\]
\[ \left( 2a - 1 \right)^2 < 0 \text { cannot be possible } . \]
\[ \therefore \left( 2a - 1 \right)^2 = 0\]
\[ \Rightarrow a = \frac{1}{2}\]
APPEARS IN
संबंधित प्रश्न
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
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).
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Show that f(x) = x – cos x is increasing for all x.
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
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
Which of the following functions is decreasing on `(0, pi/2)`?
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
The function f (x) = x2, for all real x, is ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
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 ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
