Advertisements
Advertisements
प्रश्न
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Advertisements
उत्तर
The given function is:
f(x) = 4x3 - 18x2 + 27x - 7
On differentiating both sides with respect to x, we get
f'(x) = 12x2 - 36x + 27
⇒f'(x) = 3(4x2 - 12x + 9)
⇒f'(x) = 3(2x - 3)2
which is always positive for all x ∈ R.
Since, f'(x) ≥ 0 ∀ x ∈ R,
Therefore, f(x) is always increasing on R
APPEARS IN
संबंधित प्रश्न
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Find the interval in which the following function are increasing or decreasing f(x) = 5x3 − 15x2 − 120x + 3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Function f(x) = loga x is increasing on R, if
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.
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.
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) 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 ______
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Function given by f(x) = sin x is strictly increasing in.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
A function f is said to be increasing at a point c if ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
