Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 2 x^3 - 24x + 107\]
\[f'\left( x \right) = 6 x^2 - 24 = 6 \left( x^2 - 4 \right) = 6 \left( x + 2 \right)\left( x - 2 \right)\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 6 \left( x + 2 \right)\left( x - 2 \right) > 0\]
\[ \Rightarrow \left( x + 2 \right)\left( x - 2 \right) > 0 \left[ \text { Since }6 > 0, 6 \left( x + 2 \right)\left( x - 2 \right) > 0 \Rightarrow \left( x + 2 \right)\left( x - 2 \right) > 0 \right]\]
\[ \Rightarrow x < - 2 \ or \ x > 2\]
\[ \Rightarrow x \in \left( - \infty , - 2 \right) \cup \left( 2, \infty \right)\]
\[\text { So },f(x)\text { is increasing on } x \in \left( - \infty , - 2 \right) \cup \left( 2, \infty \right).\]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 6 \left( x + 2 \right)\left( x - 2 \right) < 0\]
\[ \Rightarrow \left( x + 2 \right)\left( x - 2 \right) < 0 \left[ \text { Since } 6 > 0, 6 \left( x + 2 \right)\left( x - 2 \right) < 0 \Rightarrow \left( x + 2 \right)\left( x - 2 \right) < 0 \right]\]
\[ \Rightarrow - 2 < x < 2 \]
\[ \Rightarrow x \in \left( - 2, 2 \right)\]
\[\text { So },f(x)\text { is decreasing on }x \in \left( - 2, 2 \right) .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
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.
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 − 9x − x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Write the set of values of 'a' for which f(x) = loga x is increasing 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 interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
Every invertible function is
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
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.
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
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 case of decreasing functions, slope of tangent and hence derivative is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
