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 given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing 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) = 5 + 36x + 3x2 − 2x3 ?
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) = (x − 1) (x − 2)2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
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) = x3 − 6x2 + 9x + 15 ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
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 ______.
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 ______
The function f(x) = x3 - 3x is ______.
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
The function f (x) = 2 – 3 x is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
If f(x) = x + cosx – a then ______.
