Advertisements
Advertisements
Question
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
Options
a2 − 3b − 15 > 0
a2 − 3b + 15 > 0
a2 − 3b + 15 < 0
a > 0 and b > 0
Advertisements
Solution
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
RELATED QUESTIONS
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
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)`
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
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 − 12x2 + 18x + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)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 the function f given by f(x) = 10x is increasing for all x ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
The function f(x) = cot−1 x + x increases in the interval
Let f(x) = x3 − 6x2 + 15x + 3. Then,
The function f(x) = x2 e−x is monotonic increasing when
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
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
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) = x3 – 6x2 – 36x + 7 is strictly increasing
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing 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
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
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)`.
y = log x satisfies for x > 1, the inequality ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] increasing at \[x_0\]?
Let \[f\] be continuous on \[[a,b]\] and differentiable on \[(a,b)\]. If \[f'(x)\leq0\] for every \[x\in(a,b)\], what follows?
In the procedure for finding intervals, which conclusion is correct when \[f'(x)<0\]?
For \[f(x)=x^3-3x^2+4x\], what is \[f'(x)\]?
The critical points \[x=-2\] and \[x=3\] divide the real line into which three intervals?
