Advertisements
Advertisements
प्रश्न
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
Advertisements
उत्तर
\[f\left( x \right) = \cos x + a^2 x + b\]
\[f'\left( x \right) = a^2 - \sin x\]
\[\text { Given :f(x) is strictly increasing on R }.\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in R\]
\[ \Rightarrow a^2 - \sin x > 0, \forall x \in R\]
\[ \Rightarrow a^2 > \sin x, \forall x \in R\]
\[\text { We know that the maximum value of sin x is 1 }.\]
\[\text { Since } a^2 > \sin x, a^2\text { is always greater than 1 }.\]
\[ \Rightarrow a^2 > 1\]
\[ \Rightarrow a^2 - 1 > 0\]
\[ \Rightarrow \left( a + 1 \right)\left( a - 1 \right) > 0\]
\[ \Rightarrow a \in ( - \infty , - 1) \cup (1, \infty )\]
APPEARS IN
संबंधित प्रश्न
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
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 function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
Find the interval in which the following function are increasing or decreasing f(x) = x2 + 2x − 5 ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
The function f(x) = xx decreases on the interval
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
Find `dy/dx,if e^x+e^y=e^(x-y)`
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly 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
The function f(x) = 9 - x5 - x7 is decreasing for
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
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) = tan-1 x is ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
Function given by f(x) = sin x is strictly increasing in.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
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 ______.
A function f is said to be increasing at a point c if ______.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] increasing at \[x_0\]?
If \[f'(x)<0\] throughout an interval, then \[f\] is
For \[f'(x)=12(x-3)(x+2)\], what are the critical points obtained from \[f'(x)=0\]?
