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 of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
The interval in which y = x2 e–x is increasing is ______.
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 12x2 + 18x + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
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).
What are the values of 'a' for which f(x) = ax is increasing on R ?
The function f(x) = xx decreases on the interval
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
Function f(x) = ax is increasing on R, if
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], 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`
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Show that f(x) = x – cos x is increasing for all x.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
The function f(x) = 9 - x5 - x7 is decreasing for
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 ______
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
