Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x3 ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 5 + 36x + 3 x^2 - 2 x^3 \]
\[f'\left( x \right) = 36 + 6x - 6 x^2 \]
\[ = - 6 \left( x^2 - x - 6 \right)\]
\[ = - 6 \left( x - 3 \right)\left( x + 2 \right)\]
\[\text{ For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow - 6 \left( x - 3 \right)\left( x + 2 \right) > 0 \]
\[ \Rightarrow \left( x - 3 \right)\left( x + 2 \right) < 0 \left[ \text {Since} - 6 < 0, - 6 \left( x - 1 \right)\left( x + 2 \right) > 0 \Rightarrow \left( x - 1 \right)\left( x + 2 \right) < 0 \right]\]
\[ \Rightarrow - 2 < x < 3 \]
\[ \Rightarrow x \in \left( - 2, 3 \right)\]
\[\text { So },f(x)\text { is increasing on } \left( - 2, 3 \right) . \]
\[\text { For }f(x) \text { to be decreasing, we must have}\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow - 6 \left( x - 3 \right)\left( x + 2 \right) < 0\]
\[ \Rightarrow \left( x - 3 \right)\left( x + 2 \right) > 0 \left[ \text { Since } - 6 < 0, - 6 \left( x - 1 \right)\left( x + 2 \right) < 0 \Rightarrow \left( x - 1 \right)\left( x + 2 \right) > 0 \right]\]
\[ \Rightarrow x < - 2 \ or \ x > 3 \]
\[ \Rightarrow x \in \left( - \infty , - 2 \right) \cup \left( 3, \infty \right)\]
\[\text { So,}f(x)\text { is decreasing on } \left( - \infty , - 2 \right) \cup \left( 3, \infty \right) .\]
APPEARS IN
संबंधित प्रश्न
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
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 − 9x2 − 12x + 1 ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Function f(x) = cos x − 2 λ x is monotonic decreasing when
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
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. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
The slope of tangent at any point (a, b) is also called as ______.
The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing is ______
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
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?
The function `1/(1 + x^2)` is increasing in the interval ______
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
Which of the following functions is decreasing on `(0, pi/2)`?
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
y = log x satisfies for x > 1, the inequality ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
