Advertisements
Advertisements
Question
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
Advertisements
Solution
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[ f\left( x \right) = 2 x^3 - 15 x^2 + 36x + 1\]
\[f'\left( x \right) = 6 x^2 - 30x + 36\]
\[ = 6 \left( x^2 - 5x + 6 \right)\]
\[ = 6 \left( x - 2 \right)\left( x - 3 \right)\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 6 \left( x - 2 \right)\left( x - 3 \right) > 0\]
\[ \Rightarrow \left( x - 2 \right)\left( x - 3 \right) > 0 \left[ \text { Since } 6 > 0, 6\left( x - 2 \right)\left( x - 3 \right) > 0 \Rightarrow \left( x - 2 \right)\left( x - 3 \right) > 0 \right]\]
\[ \Rightarrow x < 2 \ or \ x > 3\]
\[ \Rightarrow x \in \left( - \infty , 2 \right) \cup \left( 3, \infty \right)\]
\[\text { So },f(x)\text { is increasing on } x \in \left( - \infty , 2 \right) \cup \left( 3, \infty \right).\]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 6 \left( x - 2 \right)\left( x - 3 \right) < 0\]
\[ \Rightarrow \left( x - 2 \right)\left( x - 3 \right) < 0 \left[ \text { Since } 6 > 0, 6\left( x - 2 \right)\left( x - 3 \right) < 0 \Rightarrow \left( x - 2 \right)\left( x - 3 \right) < 0 \right]\]
\[ \Rightarrow 2 < x < 3 \]
\[ \Rightarrow x \in \left( 2, 3 \right)\]
\[\text { So },f(x)\text { is decreasing on } x \in \left( 2, 3 \right) .\]
APPEARS IN
RELATED QUESTIONS
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
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 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
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) ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
The function f(x) = x2 e−x is monotonic increasing when
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 `dy/dx,if e^x+e^y=e^(x-y)`
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).
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).
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
The slope of tangent at any point (a, b) is also called as ______.
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 area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
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 ______
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
If f(x) = x3 – 15x2 + 84x – 17, then ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
If f(x) = x + cosx – a then ______.
y = log x satisfies for x > 1, the inequality ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
As one moves from left to right on a graph, what do constant \[y\]-values indicate?
Which form shows that \[f'(x)=3x^2-6x+4\] is positive for every \[x\in\mathbf{R}\]?
