Advertisements
Advertisements
प्रश्न
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
Advertisements
उत्तर
y = `x^4 – (4x^3)/3`
⇒ `"dy"/"dx"` = 4x3 – 4x2
= 4x2(x – 1)
Now, `"dy"/"dx"` = 0
⇒ x = 0, x = 1.
Since f′(x) < 0 ∀ ∈x `(- oo, 0)` ∪ (0, 1) and f is continuous in `(- oo, 0]` and [0, 1].
Therefore f is decreasing in `(- oo, 1]` and f is increasing in `[1, oo)`.
Note: Here f is strictly decreasing in `(- oo, 0)` ∪ (0, 1) and is strictly increasing in `(1, oo)`.
APPEARS IN
संबंधित प्रश्न
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.
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
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) = 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) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
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 the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Let f defined on [0, 1] be twice differentiable such that | f (x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
Every invertible function is
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
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.
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
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 ______
For every value of x, the function f(x) = `1/7^x` is ______
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function f(x) = tan-1 x is ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Function given by f(x) = sin x is strictly increasing in.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
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).
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
