Advertisements
Advertisements
प्रश्न
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
विकल्प
a2 − 3b − 15 > 0
a2 − 3b + 15 > 0
a2 − 3b + 15 < 0
a > 0 and b > 0
Advertisements
उत्तर
a2 − 3b + 15 < 0
Explanation:
\[f\left( x \right) = x^3 + a x^2 + bx + 5 \sin^2 x\]
\[f'\left( x \right) = 3 x^2 + 2ax + \left( b + 5 \sin 2x \right)\]
\[\text {Given}:f\left( x \right)\text { is increasing on R }.\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in R\]
\[ \Rightarrow 3 x^2 + 2ax + \left( b + 5 \sin 2x \right) > 0, \forall x \in R \]
\[\text { Since this quadratic function is >0, its discriminant is } <0.\]
\[ \Rightarrow \left( 2a \right)^2 - 4\left( 3 \right)\left( b + 5 \sin 2x \right) < 0\]
\[ \Rightarrow 4 a^2 - 12b - 60 \sin 2x < 0\]
\[ \Rightarrow a^2 - 3b - 15 \sin 2x < 0\]
\[\text { We know that the minimum value of sin 2x is−1}.\]
\[\therefore a^2 - 3b + 15 < 0 \]
APPEARS IN
संबंधित प्रश्न
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
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(x) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing f(x) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
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
Function f(x) = cos x − 2 λ x is monotonic decreasing when
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
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 func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
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 ______.
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
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 + cosx – a then ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] decreasing at \[x_0\]?
Which statement about \[f'(x)=0\] and constant behaviour is correct?
