Advertisements
Advertisements
प्रश्न
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Advertisements
उत्तर
\[f\left( x \right) = x^3 - 15 x^2 + 75x - 50\]
\[f'\left( x \right) = 3 x^2 - 30x + 75\]
\[ = 3 \left( x^2 - 10x + 25 \right)\]
\[ = 3 \left( x - 5 \right)^2 > 0, \forall x \in R \left[ \because \text { Square of any function is always greater than zero } \right]\]
\[\text{ So,f(x)is an increasing function for all x} \in R.\]
APPEARS IN
संबंधित प्रश्न
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
The function f(x) = x2 e−x is monotonic increasing when
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Function f(x) = ax is increasing on R, if
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
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.
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
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 values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Show that f(x) = x – cos x is increasing for all x.
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly 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 ______
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 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 ______
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
Which of the following graph represent the strictly increasing function.
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`
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing 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)
The function f(x) = x3 + 3x is increasing in interval ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
