Advertisements
Advertisements
Question
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Advertisements
Solution
We have f (x) = x2 + ax + 1
= f' (x) = 2x + a
If 1 < x < 2
= 2 < 2x < 4
= 2 + a < 2x + a < 4 + a
= 2 + a < f' (x) < 4 + a
Now f (x) is strictly increasing on (1, 2) only if f' (x) > 0 for 1 < x < 2
= 2 + a ≥ 0
= a ≥ -2
∴ Required least value of a is -2
APPEARS IN
RELATED QUESTIONS
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Prove that the logarithmic function is strictly increasing on (0, ∞).
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) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
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 \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
The function f(x) = xx decreases on the interval
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Function f(x) = ax is increasing on R, if
Function f(x) = loga x is increasing on R, if
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
The function f(x) = x9 + 3x7 + 64 is increasing on
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).
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
The slope of tangent at any point (a, b) is also called as ______.
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
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 ______.
2x3 - 6x + 5 is an increasing function, if ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
y = log x satisfies for x > 1, the inequality ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
The function f(x) = sin4x + cos4x is an increasing function if ______.
