Advertisements
Advertisements
प्रश्न
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Advertisements
उत्तर
f(x) = 2x3 – 3x2 – 12x + 6
∴ f'(x) = `d/dx(2x^3 - 3x^2 - 12x + 6)`
= 2 x 3x2 – 3 x 2x – 12 x 1 + 0
= 6x2 – 6x – 12
= 6(x2 – x – 2)
f is strictly increasing if f'(x) > 0
i.e. if 6(x2 – x – 2) > 0
i.e. if x2 – x – 2 > 0
i.e. if x2 – x > 2
i.e. if `x^2 - x + (1)/(4) > 2 + (1)/(4)`
i.e. if `(x - 1/2)^2 > (9)/(4)`
i.e. if `x - (1)/(2) > (3)/(2) or x - (1)/(2) < - (3)/(2)`
i.e. if x > 2 or x < – 1
∴ f is strictly increasing if x < – 1 or x > 2.
APPEARS IN
संबंधित प्रश्न
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
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.
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
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 f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 − 9x − x2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
Find `dy/dx,if e^x+e^y=e^(x-y)`
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
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) = x3 – 6x2 + 12x – 16, x ∈ R.
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Show that f(x) = x – cos x is increasing for all x.
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
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 ______.
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
The function f(x) = x3 - 3x is ______.
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
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) = 2 – 3 x is ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
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 ______.
