Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\]
\[f'\left( x \right) = 12 x^3 - 12 x^2 - 24x\]
\[ = 12x\left( x^2 - x - 2 \right)\]
\[ = 12x\left( x + 1 \right)\left( x - 2 \right)\]
\[\text { Here }, x = 0,x = - 1\text { and x = 2 are the critical points }.\]
\[\text { The possible intervals are }\left( - \infty , - 1 \right),\left( - 1, 0 \right),\left( 0, 2 \right)\text { and }\left( 2, \infty \right). .....(1)\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 12x\left( x + 1 \right)\left( x - 2 \right) > 0 \left[ \text { Since }, 12 > 0, 12x\left( x + 1 \right)\left( x - 2 \right) > 0 \Rightarrow x\left( x + 1 \right)\left( x - 2 \right) > 0 \right]\]
\[ \Rightarrow x\left( x + 1 \right)\left( x - 2 \right) > 0\]
\[ \Rightarrow x \in \left( - 1, 0 \right) \cup \left( 2, \infty \right) \left[ \text { From eq. } (1) \right]\]
\[\text { So,f(x)is increasing on x } \in \left( - 1, 0 \right) \cup \left( 2, \infty \right) .\]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 12x\left( x + 1 \right)\left( x - 2 \right) < 0 \left[ \text { Since }12 > 0, 12x\left( x + 1 \right)\left( x - 2 \right) < 0 \Rightarrow x\left( x + 1 \right)\left( x - 2 \right) < 0 \right]\]
\[ \Rightarrow x \left( x + 1 \right)\left( x - 2 \right) < 0\]
\[ \Rightarrow x \in \left( - \infty , - 1 \right) \cup \left( 0, 2 \right) \left[ \text { From eq. } (1) \right]\]
\[\text { So,f(x)is decreasing on x }\in \left( - \infty , - 1 \right) \cup \left( 0, 2 \right) .\]
APPEARS IN
संबंधित प्रश्न
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 function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Prove that the logarithmic function is strictly increasing on (0, ∞).
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?
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 \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
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) ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
Show that f(x) = x – cos x is increasing for all x.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
The slope of tangent at any point (a, b) is also called as ______.
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
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 ______
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
y = x(x – 3)2 decreases for the values of x given by : ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
Function given by f(x) = sin x is strictly increasing in.
y = log x satisfies for x > 1, the inequality ______.
Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.
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)
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
The function f(x) = xex(1 − x), x ∈ R, is ______.
