Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 2 x^3 - 9 x^2 + 12x - 5\]
\[f'\left( x \right) = 6 x^2 - 18x + 12\]
\[ = 6 \left( x^2 - 3x + 2 \right)\]
\[ = 6 \left( x - 1 \right)\left( x - 2 \right)\]
\[\text{ For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 6 \left( x - 1 \right)\left( x - 2 \right) > 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 2 \right) > 0 \left[ \text { Since } 6 > 0, 6 \left( x - 1 \right)\left( x - 2 \right) > 0 \Rightarrow \left( x - 1 \right)\left( x - 2 \right) > 0 \right]\]
\[ \Rightarrow x < 1 orx > 2\]
\[ \Rightarrow x \in \left( - \infty , 1 \right) \cup \left( 2, \infty \right)\]
\[\text { So },f(x)\text { is increasing on } x \in \left( - \infty , 1 \right) \cup \left( 2, \infty \right).\]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 6 \left( x - 1 \right)\left( x - 2 \right) < 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 2 \right) < 0 \left[ \text { Since } 6 > 0, 6 \left( x - 1 \right)\left( x - 2 \right) < 0 \Rightarrow \left( x - 1 \right)\left( x - 2 \right) < 0 \right]\]
\[ \Rightarrow 1 < x < 2\]
\[ \Rightarrow x \in \left( 1, 2 \right)\]
\[\text { So} ,f(x)\text { is decreasing on } x \in \left( 1, 2 \right) .\]
APPEARS IN
संबंधित प्रश्न
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
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)
Prove that the logarithmic function is strictly increasing on (0, ∞).
Find the intervals in which the function f given by `f(x) = (4sin x - 2x - x cos x)/(2 + cos x)` is (i) increasing (ii) decreasing.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
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(x) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
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(x) = x8 + 6x2 ?
Show that f(x) = e2x is increasing on R.
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
Let f(x) = x3 − 6x2 + 15x + 3. Then,
Function f(x) = | x | − | x − 1 | is monotonically increasing when
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
Function f(x) = loga x is increasing on R, if
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
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 interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] increasing at \[x_0\]?
After the domain has been divided into intervals, which quantity must be determined in each interval?
For \[f(x)=x^3-3x^2+4x\], what is \[f'(x)\]?
Which statement about \[f'(x)=0\] and constant behaviour is correct?
