Advertisements
Advertisements
प्रश्न
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Advertisements
उत्तर १
`f(x) = x^2 - 3x`
i) `f(-sqrt3) = (-sqrt3)^3 - 3(-sqrt3) = -3sqrt3 + 3sqrt3 = 0`
f(0) = 0
Also f(x) = continuos in `[-sqrt3, 0]` and differentiable in `(-sqrt3,0)`
f'(c) = 0
`=> 3x^2 - 3 = 0`
`:. 3c^2 - 3 = 0`
`c^2 = 1`
c = ±1
⇒ c = -1
उत्तर २
The given function is f(x) = x3 – 3x.
Since a polynomial function is everywhere continous and differentiable, therefore f(x) is continous on [`-sqrt3`, 0] and differentaible on (`-sqrt3`,0)
Also `f(-sqrt3) = (-sqrt3)^3 - 3(-sqrt3) = -3sqrt3 + 3sqrt3 = 0`
f(0) = (0)3 – 3 × 0 = 0
Since all the three conditions of Rolle’s theorem are satisfied, so there exists a point c ∈ (`-sqrt3,0`) such that f'(c) = 0
f(x) = x3 − 3x
f'(x) = 3x2 − 3
∴ f'(c) = 0
⇒3c2 − 3 = 0
⇒c2 − 1 = 0
⇒ (c + 1)(c − 1) = 0
⇒ c = −1 or c = 1
Now, `c != 1` [∵ 1 ∉ (`-sqrt3,0`)]
∴ c = -1, where c ∈ (`-sqrt3,0`)
Thus, the required value of c is –1.
संबंधित प्रश्न
Prove that the logarithmic function is strictly increasing on (0, ∞).
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
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) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
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 f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
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 ?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
Function f(x) = | x | − | x − 1 | is monotonically increasing when
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
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).
Show that f(x) = x – cos x is increasing for all x.
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
A ladder 20 ft Jong leans against a vertical wall. The top-end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is ______
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.
