Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20 ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 2 x^3 + 9 x^2 + 12x + 20\]
\[f'\left( x \right) = 6 x^2 + 18x + 12\]
\[ = 6 \left( x^2 + 3x + 2 \right)\]
\[ = 6 \left( x + 1 \right)\left( x + 2 \right)\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 6 \left( x + 1 \right)\left( x + 2 \right) > 0\]
\[ \Rightarrow \left( x + 1 \right)\left( x + 2 \right) > 0 \left[ \text { Since } 6 > 0, 6 \left( x + 1 \right)\left( x + 2 \right) > 0 \Rightarrow \left( x + 1 \right)\left( x + 2 \right) > 0 \right]\]
\[ \Rightarrow x < - 2 \ or \ x > - 1\]
\[ \Rightarrow x \in \left( - \infty , - 2 \right) \cup \left( - 1, \infty \right)\]
\[\text { So },f(x)\text { is increasing on } x \in \left( - \infty , - 2 \right) \cup \left( - 1, \infty \right).\]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 6 \left( x + 1 \right)\left( x + 2 \right) < 0\]
\[ \Rightarrow \left( x + 1 \right)\left( x + 2 \right) < 0 \left[ \text { Since } 6 > 0, 6 \left( x + 1 \right)\left( x + 2 \right) < 0 \Rightarrow \left( x + 1 \right)\left( x + 2 \right) < 0 \right]\]
\[ \Rightarrow - 2 < x < - 1 \]
\[ \Rightarrow x \in \left( - 2, - 1 \right)\]
\[\text { So, }f(x)\text { is decreasing on }x \in \left( - 2, - 1 \right) .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
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
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = x2 + 2x − 5 ?
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) = 5x3 − 15x2 − 120x + 3 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing 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.
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Show that f(x) = x – cos x is increasing for all x.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
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) = 2x3 – 6x2 + 6x + 24 is strictly increasing
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
The function f(x) = sin x + 2x is ______
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
The function `1/(1 + x^2)` is increasing in the interval ______
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
y = x(x – 3)2 decreases for the values of x given by : ______.
The function f(x) = tan-1 x is ____________.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
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)
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] increasing at \[x_0\]?
Let \[f\] be continuous on \[[a,b]\] and differentiable on \[(a,b)\]. If \[f'(x)\geq0\] for every \[x\in(a,b)\], what follows?
As one moves from left to right on a graph, what does a decrease in the \[y\]-values indicate?
For \[f(x)=x^3-3x^2+4x\], what is \[f'(x)\]?
The function \[f(x)=x^3-3x^2+4x\], \[x\in\mathbf{R}\], is
