Advertisements
Advertisements
Question
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Advertisements
Solution
\[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\]
\[f'\left( x \right) = 12 x^2 - 36x + 27\]
\[ \Rightarrow f'\left( x \right) = 3\left( 4 x^2 - 12x + 9 \right)\]
\[ \Rightarrow f'\left( x \right) = 3 \left( 2x - 3 \right)^2 > 0, \forall x \in R\]
So, f(x) is increasing on R.
APPEARS IN
RELATED QUESTIONS
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
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:
(x + 1)3 (x − 3)3
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
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) = x2 + 2x − 5 ?
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) = 5x3 − 15x2 − 120x + 3 ?
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) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
Every invertible function is
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
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.
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 y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
The function f(x) = 9 - x5 - x7 is decreasing for
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
The function f(x) = tan-1 x is ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
