Advertisements
Advertisements
Question
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
Advertisements
Solution
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = x^3 - 12 x^2 + 36x + 17\]
\[f'\left( x \right) = 3 x^2 - 24x + 36\]
\[ = 3 \left( x^2 - 8x + 12 \right)\]
\[ = 3 \left( x - 2 \right)\left( x - 6 \right)\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 3 \left( x - 2 \right)\left( x - 6 \right) > 0\]
\[ \Rightarrow \left( x - 2 \right)\left( x - 6 \right) > 0 \left[ \text { Since } 3 > 0, 3 \left( x - 2 \right)\left( x - 6 \right) > 0 \Rightarrow \left( x - 2 \right)\left( x - 6 \right) > 0 \right]\]
⇒ x < 2 or x > 6
\[ \Rightarrow x \in \left( - \infty , 2 \right) \cup \left( 6, \infty \right)\]
\[\text { So },f(x)\text { is increasing on } x \in \left( - \infty , 2 \right) \cup \left( 6, \infty \right).\]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 3 \left( x - 2 \right)\left( x - 6 \right) < 0\]
\[ \Rightarrow \left( x - 2 \right)\left( x - 6 \right) < 0 \left[ \text { Since } 3 > 0, 3 \left( x - 2 \right)\left( x - 6 \right) < 0 \Rightarrow \left( x - 2 \right)\left( x - 6 \right) < 0 \right]\]
\[ \Rightarrow 2 < x < 6 \]
\[ \Rightarrow x \in \left( 2, 6 \right)\]
\[\text { So,}f(x)\text { is decreasing on } x \in \left( 2, 6 \right) .\]
APPEARS IN
RELATED QUESTIONS
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Prove that the logarithmic function is strictly increasing on (0, ∞).
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
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) = 5x3 − 15x2 − 120x + 3 ?
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
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) = sin x is an increasing function on (−π/2, π/2) ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
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) = x + cos x − a is an increasing function on R for all values of a ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
The function f(x) = cot−1 x + x increases in the interval
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Let f(x) = x3 − 6x2 + 15x + 3. Then,
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
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
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
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 ______.
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
For every value of x, the function f(x) = `1/7^x` is ______
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
