English

If the Function F(X) = 2 Tan X + (2a + 1) Loge | Sec X | + (A − 2) X Is Increasing on R, Then - Mathematics

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

MCQ
Advertisements

Solution

\[f(x) = 2 \tan x + \left( 2a + 1 \right) \log_e \left| \sec x \right| + \left( a - 2 \right) x\]

\[\text { When }\sec x > 0 \Rightarrow \left| \sec x \right| = \sec x\]

\[\frac{d}{dx}\left\{ f\left( x \right) \right\} = 2 \sec^2 x + \left( 2a + 1 \right)\frac{1}{\sec x} \times \sec x \tan x + \left( a - 2 \right) \]

\[ = 2 \sec^2 x + \left( 2a + 1 \right)\tan x + \left( a - 2 \right) \]

\[\text { For  f(x) to be increasing}, \]

\[2se c^2 x + \left( 2a + 1 \right)\tan x + \left( a - 2 \right) \geqslant 0\]

\[ \Rightarrow 2 + 2 \tan^2 x + \left( 2a + 1 \right)\tan x + a - 2 \geqslant 0\]

\[ \Rightarrow 2 \tan^2 x + \left( 2a + 1 \right)\tan x + a \geqslant 0\]

\[\text  { Its discriminant } \leqslant 0 \left[ \because a x^2 + bx + c \geqslant 0 \Rightarrow b^2 - 4ac \leqslant 0 \right]\]

\[ \Rightarrow \left( 2a + 1 \right)^2 - 4 . 2 . a \leqslant 0\]

\[ \Rightarrow 4 a^2 - 4a + 1 \leqslant 0\]

\[ \Rightarrow \left( 2a - 1 \right)^2 \leqslant 0\]

\[ \left( 2a - 1 \right)^2 < 0 \text { cannot be possible } . \]

\[ \therefore \left( 2a - 1 \right)^2 = 0\]

\[ \Rightarrow a = \frac{1}{2}\]

\[\text { When } \sec x < 0 \Rightarrow \left| \sec x \right| = - \sec x\]

\[\frac{d}{dx}\left\{ f\left( x \right) \right\} = 2 \sec^2 x + \left( 2a + 1 \right)\frac{1}{- \sec x} \times \sec x \tan x + \left( a - 2 \right)\]

\[ = 2 \sec^2 x - \left( 2a + 1 \right)\tan x + \left( a - 2 \right) \]

\[\text { For f(x) to be increasing,} \]

\[2se c^2 x - \left( 2a + 1 \right)\tan x + \left( a - 2 \right) \geqslant 0\]

\[ \Rightarrow 2 + 2 \tan^2 x - \left( 2a + 1 \right)\tan x + a - 2 \geqslant 0\]

\[ \Rightarrow 2 \tan^2 x - \left( 2a + 1 \right)\tan x + a \geqslant 0 \]

\[\text { Its discriminant } \leqslant 0 \left[ \because a x^2 + bx + c \geqslant 0 \Rightarrow b^2 - 4ac \leqslant 0 \right]\]

\[ \Rightarrow \left\{ - \left( 2a + 1 \right) \right\}^2 - 4 . 2 . a \leqslant 0\]

\[ \Rightarrow 4 a^2 - 4a + 1 \leqslant 0\]

\[ \Rightarrow \left( 2a - 1 \right)^2 \leqslant 0\]

\[ \left( 2a - 1 \right)^2 < 0 \text { cannot be possible } . \]

\[ \therefore \left( 2a - 1 \right)^2 = 0\]

\[ \Rightarrow a = \frac{1}{2}\]

shaalaa.com
  Is there an error in this question or solution?
Chapter 17: Increasing and Decreasing Functions - Exercise 17.4 [Page 40]

APPEARS IN

RD Sharma Mathematics [English] Class 12
Chapter 17 Increasing and Decreasing Functions
Exercise 17.4 | Q 8 | Page 40

RELATED QUESTIONS

Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing


The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.

(A) increasing

(B) decreasing

(C) increasing and decreasing

(D) neither increasing nor decreasing


Find the intervals in which the following functions are strictly increasing or decreasing:

 (x + 1)3 (x − 3)3


Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).


Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?


Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x?


Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?


Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?


Show that f(x) = x − sin x is increasing for all x ∈ R ?


Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?


Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?


Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?


Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?


Find `dy/dx,if e^x+e^y=e^(x-y)`


 Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R. 


Find the intervals in which function f given by f(x)  = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .


Find the value of x, such that f(x) is decreasing function.

f(x) = 2x3 - 15x2 - 144x - 7 


Choose the correct alternative:

The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is


If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______


If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.


Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`


The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.


The function f (x) = 2 – 3 x is ____________.


Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.


Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.


`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.


The function `"f"("x") = "x"/"logx"` increases on the interval


Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 –  h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.


Show that function f(x) = tan x is increasing in `(0, π/2)`.


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).


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 ______.


The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.


A function f is said to be increasing at a point c if ______.


The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.


Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.


The function f(x) = sin4x + cos4x is an increasing function if ______.


Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×