Advertisements
Advertisements
Question
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?
Advertisements
Solution
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = x^3 - 6 x^2 - 36x + 2\]
\[f'\left( x \right) = 3 x^2 - 12x - 36\]
\[ = 3 \left( x^2 - 4x - 12 \right)\]
\[ = 3 \left( x - 6 \right)\left( x + 2 \right)\]
\[\text { For }f(x) \text { to be increasing, we must have },\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 3 \left( x - 6 \right)\left( x + 2 \right) > 0\]
\[ \Rightarrow \left( x - 6 \right)\left( x + 2 \right) > 0 \left[ \text { Since } 3 > 0, 3 \left( x - 6 \right)\left( x + 2 \right) > 0 \Rightarrow \left( x - 6 \right)\left( x + 2 \right) > 0 \right]\]
\[ \Rightarrow x < - 2 \ or \ x > 6\]
\[ \Rightarrow x \in \left( - \infty , - 2 \right) \cup \left( 6, \infty \right)\]
\[\text { So,}f(x)\text { is increasing on } x \in \left( - \infty , - 2 \right) \cup \left( 6, \infty \right).\]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 3 \left( x - 6 \right)\left( x + 2 \right) < 0\]
\[ \Rightarrow \left( x - 6 \right)\left( x + 2 \right) < 0 \left[ \text { Since } 3 > 0, 3 \left( x - 6 \right)\left( x + 2 \right) < 0 \Rightarrow \left( x - 6 \right)\left( x + 2 \right) < 0 \right]\]
\[ \Rightarrow - 2 < x < 6 \]
\[ \Rightarrow x \in \left( - 2, 6 \right)\]
\[\text{ So },f(x)\text { is decreasing on } x \in \left( - 2, 6 \right) .\]
APPEARS IN
RELATED QUESTIONS
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
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:
−2x3 − 9x2 − 12x + 1
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 − 12x2 + 18x + 15 ?
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) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
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 ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Every invertible function is
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
The slope of tangent at any point (a, b) is also called as ______.
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
The function f(x) = sin x + 2x is ______
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` 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 ____________.
Which of the following graph represent the strictly increasing function.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
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 ______.
Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
