Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 12x2 + 18x + 15 ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 2 x^3 - 12 x^2 + 18x + 15\]
\[f'\left( x \right) = 6 x^2 - 24x + 18\]
\[ = 6 \left( x^2 - 4x + 3 \right)\]
\[ = 6 \left( x - 1 \right)\left( x - 3 \right)\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 6 \left( x - 1 \right)\left( x - 3 \right) > 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 3 \right) > 0 \left[ \text { Since } 6 > 0, 6 \left( x - 1 \right)\left( x - 3 \right) > 0 \Rightarrow \left( x - 1 \right)\left( x - 3 \right) > 0 \right]\]
\[ \Rightarrow x < 1 \ or \ x > 3\]
\[ \Rightarrow x \in \left( - \infty , 1 \right) \cup \left( 3, \infty \right)\]
\[\text { So },f(x)\text { is increasing on }\left( - \infty , 1 \right) \cup \left( 3, \infty \right) . \]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 6 \left( x - 1 \right)\left( x - 3 \right) < 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 3 \right) < 0 \left[ \text { Since} 6 > 0, 6 \left( x - 1 \right)\left( x - 3 \right) < 0 \Rightarrow \left( x - 1 \right)\left( x - 3 \right) < 0 \right]\]
\[ \Rightarrow 1 < x < 3 \]
\[ \Rightarrow x \in \left( 1, 3 \right)\]
\[\text { So },f(x)\text { is decreasing on }\left( 1, 3 \right).\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on 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 \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong is
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(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 ______
The function f(x) = sin x + 2x is ______
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The function f(x) = tanx – x ______.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, 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.
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 ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
As one moves from left to right on a graph, what does an increase in the \[y\]-values indicate?
The function \[f(x)=x^3-3x^2+4x\], \[x\in\mathbf{R}\], is
For \[f(x)=4x^3-6x^2-72x+30\], on which intervals is \[f\] increasing?
