Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2 ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with } a < b, x < a \text { or } x>b.\].
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f(x) = 10 - 6x - 2 x^2 \]
\[f'(x) = - 6 - 4x\]
\[\text { For } f(x) \text { to be increasing, we must have } \]
\[f'(x) > 0\]
\[ \Rightarrow - 6 - 4x > 0\]
\[ \Rightarrow - 4x > 6\]
\[ \Rightarrow x < \frac{- 3}{2}\]
\[ \Rightarrow x \in \left( - \infty , \frac{- 3}{2} \right)\]
\[\text { So }, f(x) \text { is increasing on } \left( - \infty , \frac{- 3}{2} \right) . \]
\[\text { For } f(x) \text { to be decreasing, we must have } \]
\[f'(x) < 0\]
\[ \Rightarrow - 6 - 4x < 0\]
\[ \Rightarrow - 4x < 6\]
\[ \Rightarrow x > \frac{- 6}{4}\]
\[ \Rightarrow x > \frac{- 3}{2}\]
\[ \Rightarrow x \in \left( \frac{- 3}{2}, \infty \right)\]
\[\text { So }, f(x) \text { is decreasing on } \left( \frac{- 3}{2}, \infty \right) .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Prove that the logarithmic function is strictly increasing on (0, ∞).
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Show that f(x) = \[\frac{1}{x}\] is a decreasing function 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 − 15x2 + 36x + 1 ?
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 ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
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 ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
The function f(x) = x2 e−x is monotonic increasing when
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Find `dy/dx,if e^x+e^y=e^(x-y)`
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
