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
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
The function f (x) = 2 – 3 x is ____________.
The function f(x) = tan-1 x is ____________.
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
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.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
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 ______.
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 ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] increasing at \[x_0\]?
Which form shows that \[f'(x)=3x^2-6x+4\] is positive for every \[x\in\mathbf{R}\]?
For \[f'(x)=12(x-3)(x+2)\], what is the nature of \[f\] on \[(-2,3)\]?
