Advertisements
Advertisements
प्रश्न
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
विकल्प
a = b
\[a = \frac{1}{2}b\]
\[a \leq - \frac{1}{2}\]
\[a > - \frac{3}{2}\]
Advertisements
उत्तर
\[a \leq - \frac{1}{2}\]
\[Given: f\left( x \right) = \cos \left| x \right| - 2ax + b\]
\[\text { Now}, \left| x \right| =\begin{cases} x ,& x \geq 0 \\ - x, & x < 0 \end{cases}\]
\[\text { and } \cos \left| x \right| = \begin{cases} \cos\left( x \right) , & x \geq 0 \\cos\left( - x \right) = cos\left( x \right), & x < 0\end{cases}\]
\[ \therefore \cos\left| x \right| = \cos x , \forall x \in R\]
\[ \therefore f\left( x \right) = \cos x - 2ax + b\]
\[ \Rightarrow f'\left( x \right) = - \sin x - 2a\]
\[\text { It is given that f(x) is increasing } . \]
\[ \Rightarrow f'\left( x \right) \geq 0\]
\[ \Rightarrow - \sin x - 2a \geq 0\]
\[ \Rightarrow \sin x + 2a \leq 0\]
\[ \Rightarrow 2a \leq - \sin x\]
\[\text { The least value of -sin x is -1 }.\]
\[ \Rightarrow 2a \leq - 1\]
\[ \Rightarrow a \leq \frac{- 1}{2}\]
APPEARS IN
संबंधित प्रश्न
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Prove that the function f given by f(x) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`
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) = 2x3 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
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
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong is
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
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 '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: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` 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 ______.
y = log x satisfies for x > 1, the inequality ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
