Advertisements
Advertisements
प्रश्न
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
Advertisements
उत्तर
f(x) = 2x3 – 6x2 + 6x + 24
∴ f′(x) = 6x2 – 12x + 6
= 6(x2 – 2x + 1)
= 6(x – 1)2
f(x) is strictly increasing, if f′(x) > 0
∴ 6(x – 1)2 > 0
∴ (x – 1)2 > 0 for all x ∈ R, x ≠ 1
Thus, f(x) is strictly increasing for x ∈ R – {1}.
APPEARS IN
संबंधित प्रश्न
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
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) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20 ?
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) = 2x3 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
The function f(x) = cot−1 x + x increases in the interval
Function f(x) = cos x − 2 λ x is monotonic decreasing when
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
Function f(x) = | x | − | x − 1 | is monotonically increasing when
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.
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).
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
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 ______
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong is
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
The function f(x) = x3 - 3x is ______.
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
Which of the following functions is decreasing on `(0, pi/2)`?
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 ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` 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) = xex(1 − x), x ∈ R, is ______.
