Advertisements
Advertisements
प्रश्न
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).
Advertisements
उत्तर
Let x1, x2, ∈ (a, b) such that x1 < x2 ∈ f (x) is differentiable on (a, b) and [x1, x2] ⊂ (a, b)
∴ f(x) is continuous on [x1, x2] and differentiable on (x1, x2).
∴ According to Lagrange mean theorem,
Here there exists c ∈ (x1, x2) such that
`f'(c) = (f(x_2) - f(x_1))/(x_2 - x_1)` ...(1)
Since for all x ∈ (a, b), f'(x) > 0
∴ In particular, f'(c) > 0
Now, f'(c) > 0 `=> (f(x_2) - f(x_1))/(x_2 - x_1) > 0`
⇒ f(x2) - f(x1) > 0 ...[∵ x2 - x1 > 0 when x1 - x2]
⇒ f(x2) > f(x1)
⇒ f(x1) < f(x2), if x1 < x2
Because x1, x2 are arbitrary points in (a, b).
∴ x1 < x2
⇒ f(x1) < f(x2) for all
x1, x2 ∈ (a, b)
∴ f(x) is increasing in (a, b).
APPEARS IN
संबंधित प्रश्न
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
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) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing f(x) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
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 ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Let f defined on [0, 1] be twice differentiable such that | f (x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?
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 ?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
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)
The function f(x) = x3 + 3x is increasing in interval ______.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] decreasing at \[x_0\]?
For \[f(x)=4x^3-6x^2-72x+30\], which factorization of \[f'(x)\] is correct?
