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
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
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).
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x3 ?
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) = x8 + 6x2 ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
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) ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Every invertible function is
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
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
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The function f(x) = tanx – x ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
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 ______.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
If f(x) = x + cosx – a then ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
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)
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] increasing at \[x_0\]?
The critical points \[x=-2\] and \[x=3\] divide the real line into which three intervals?
