Advertisements
Advertisements
Question
Find the interval in which the following function are increasing or decreasing f(x) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Solution
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[\ f\left( x \right) = 5 x^\frac{3}{2} - 3 x^\frac{5}{2} , x > 0\]
\[f'\left( x \right) = \frac{15}{2} x^\frac{1}{2} - \frac{15}{2} x^\frac{3}{2} \]
\[ = \frac{15}{2} x^\frac{1}{2} \left( 1 - x \right)\]
\[\text { Here }, 0, 1 \text { are the roots } .\]
\[\text { The possible intervals are }\left( - \infty , 0 \right),\left( 0, 1 \right)\text { and }\left( 1, \infty \right)...(1)\]
\[\text { For f(x) to be increasing, we must have}\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \frac{15}{2} x^\frac{1}{2} \left( 1 - x \right) > 0\]
\[ \Rightarrow x \in \left( 0, 1 \right)\]
\[\text { So,f(x)is increasing on } \left( 0, 1 \right) . \]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow \frac{15}{2} x^\frac{1}{2} \left( 1 - x \right) < 0\]
\[ \Rightarrow x \in \left( 1, \infty \right)\]
\[\text { So,f(x)is decreasing on }\left( 1, \infty \right).\]
APPEARS IN
RELATED QUESTIONS
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
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) = 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) = (x − 1) (x − 2)2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Show that the function f given by f(x) = 10x is increasing for all x ?
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).
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
Function f(x) = | x | − | x − 1 | is monotonically increasing when
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
Which of the following functions is decreasing on `(0, pi/2)`?
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) = x2 – 2x is increasing in the interval ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
Which of the following graph represent the strictly increasing function.
If f(x) = x + cosx – a then ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
