Advertisements
Advertisements
प्रश्न
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Advertisements
उत्तर
f(x) = `x/logx`
∴ f'(x) = `d/dx(x/logx)`
= `((logx).d/dx(x) - x*d/dx(logx))/(logx)^2`
= `((logx) xx 1 - x xx 1/x)/(logx)^2`
= `(logx - 1)/(log x)^2`
f is increasing if f'(x) ≥ 0
i.e. if `(log x - 1)/(logx)^2 ≥ 0`
i.e. if log x – 1 ≥ 0 ...[∵ (log x)2 > 0]
i.e. if log x ≥ 1
i.e. if log x ≥ log e ...[∵ log e = 1]
i.e. if x ≥ e
∴ f is increasing on `[e, oo)`
f is decreasing if f'(x) ≤ 0
i.e. if `(log x - 1)/(logx)^2 ≤ 0`
i.e. if log x – 1 ≤ 0 ...[∵ (log x)2 > 0]
i.e. if log x ≤ 1
i.e. if log x ≤ log e
i.e. if x ≤ e
Also, x > 0 and x ≠ 1 because f(x) = `x/logx` is not defined at x = 1
∴ f is decreasing in (0, e] – {1}
Hence, f is increasing in `[e, oo)` and decreasing in (0, e] – {1}.
APPEARS IN
संबंधित प्रश्न
Prove that the function f given by f(x) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
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 − 9x2 − 12x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
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 ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
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) ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
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
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Show that f(x) = x – cos x is increasing for all x.
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) 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) = x3 – 15x2 + 84x – 17, then ______.
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
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) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
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 ______.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
If f(x) = x + cosx – a then ______.
Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.
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)
