Advertisements
Advertisements
प्रश्न
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
उत्तर
We have f (x) = x2 + ax + 1
= f' (x) = 2x + a
If 1 < x < 2
= 2 < 2x < 4
= 2 + a < 2x + a < 4 + a
= 2 + a < f' (x) < 4 + a
Now f (x) is strictly increasing on (1, 2) only if f' (x) > 0 for 1 < x < 2
= 2 + a ≥ 0
= a ≥ -2
∴ Required least value of a is -2
APPEARS IN
संबंधित प्रश्न
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
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 \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
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) ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
The function f(x) = xx decreases on the interval
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
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.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Test whether the following functions are increasing or decreasing : f(x) = `(1)/x`, x ∈ R , x ≠ 0.
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
For every value of x, the function f(x) = `1/7^x` is ______
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
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 ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The function f(x) = tan-1 x is ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
If f(x) = x + cosx – a then ______.
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)
