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 ?
Advertisements
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
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
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) = x8 + 6x2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Show that f(x) = e2x is increasing on R.
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
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) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
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 y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
If f(x) = x + cosx – a then ______.
Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
