Advertisements
Advertisements
प्रश्न
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Advertisements
उत्तर
\[f\left( x \right) = x^3 - 3 x^2 + 4x\]
\[f'\left( x \right) = 3 x^2 - 6x + 4\]
\[ = 3\left( x^2 - 2x \right) + 4\]
\[ = 3\left( x^2 - 2x + 1 \right) - 3 + 4\]
\[ = 3 \left( x - 1 \right)^2 + 1 > 0, \forall x \in R\]
\[\text { Hence , f(x) is strictly increasing on R } .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
The interval in which y = x2 e–x is increasing is ______.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
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) = x4 − 4x3 + 4x2 + 15 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Every invertible function is
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Function f(x) = loga x is increasing on R, if
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.
Show that f(x) = x – cos x is increasing for all x.
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
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
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 `1/(1 + x^2)` is increasing in the interval ______
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function f(x) = x2 – 2x is increasing in the interval ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
If f(x) = x + cosx – a then ______.
A function f is said to be increasing at a point c if ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
