Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
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 f given by f(x) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
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) = 5 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Let f defined on [0, 1] be twice differentiable such that | f (x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
Function f(x) = ax is increasing on R, if
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
The slope of tangent at any point (a, b) is also called as ______.
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
A ladder 20 ft Jong leans against a vertical wall. The top-end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is ______
The function f (x) = 2 – 3 x is ____________.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
Function given by f(x) = sin x is strictly increasing in.
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(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
