Advertisements
Advertisements
Question
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?
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
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
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:
−2x3 − 9x2 − 12x + 1
Prove that the logarithmic function is strictly increasing on (0, ∞).
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
Let f(x) = x3 − 6x2 + 15x + 3. Then,
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = cos x − 2 λ x is monotonic decreasing when
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Show that f(x) = x – cos x is increasing for all x.
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
The slope of tangent at any point (a, b) is also called as ______.
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
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
The function f(x) = 9 - x5 - x7 is decreasing for
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
Function given by f(x) = sin x is strictly increasing in.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
The function f(x) = sin4x + cos4x is an increasing function if ______.
