Advertisements
Advertisements
प्रश्न
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Advertisements
उत्तर
\[f\left( x \right) = \tan x\]
\[f'\left( x \right) = \sec^2 x\]
\[\text { Here },\]
\[\frac{- \pi}{2} < x < \frac{\pi}{2}\]
\[ \Rightarrow \sec x > 0 \left[ \because Sec \text { function is positive in first and fourth quadrant } \right]\]
\[ \Rightarrow \sec^2 x > 0\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in \left( \frac{- \pi}{2}, \frac{\pi}{2} \right)\]
\[\text { So },f(x)\text { is increasing on } \left( \frac{- \pi}{2}, \frac{\pi}{2} \right) .\]
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.
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that the logarithmic function is strictly increasing on (0, ∞).
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
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) = 5x3 − 15x2 − 120x + 3 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
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).
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
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 ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
Function f(x) = loga x is increasing on R, if
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).
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
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 f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Show that f(x) = x – cos x is increasing for all x.
The slope of tangent at any point (a, b) is also called as ______.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` 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 ______.
y = log x satisfies for x > 1, the inequality ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function 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 values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
The function f(x) = xex(1 − x), x ∈ R, is ______.
