Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
We have `f (x) = x + 1/x, x in I`
Differentiating w.r.t.x, we get
`f' (x) = 1 - 1/x^2 = (x^2 - 1)/x^2`
`x^2 > 0 (1, 1), x^2 - 1 > 0 = x^2 > 1`
= `x < - 1 or x > 1`
= `x in (-oo, -1) or x in (1, oo)`
= `x in (-oo, -1) cup (1, oo) `
= `x in R - (-1, 1)`
= f (x) is strictly increasing on I
(∵ I is an interval which is a subset of R - (-1, 1))
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 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 following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
The interval in which y = x2 e–x is increasing is ______.
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
Function f(x) = cos x − 2 λ x is monotonic decreasing when
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
The function f(x) = 9 - x5 - x7 is decreasing for
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] ______
For every value of x, the function f(x) = `1/7^x` is ______
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
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 interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
The function f(x) = sin4x + cos4x is an increasing function if ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
Let \[f\] be continuous on \[[a,b]\] and differentiable on \[(a,b)\]. If \[f'(x)\leq0\] for every \[x\in(a,b)\], what follows?
As one moves from left to right on a graph, what does an increase in the \[y\]-values indicate?
