Advertisements
Advertisements
प्रश्न
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
पर्याय
(1, 2)
(2, 3)
(1, 3)
(2, 4)
Advertisements
उत्तर
(2, 3)
\[\text { Given: } \hspace{0.167em} f\left( x \right) = 2 \log \left( x - 2 \right) - x^2 + 4x + 1\]
\[\text { Domain of f }\left( x \right) is\left( 2, \infty \right).\]
\[f'\left( x \right) = \frac{2}{x - 2} - 2x + 4\]
\[ = \frac{2 - 2 x^2 + 4x + 4x - 8}{x - 2}\]
\[ = \frac{- 2 x^2 + 8x - 6}{x - 2}\]
\[ = \frac{- 2 \left( x^2 - 4x + 3 \right)}{x - 2}\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \frac{- 2 \left( x^2 - 4x + 3 \right)}{x - 2} > 0\]
\[ \Rightarrow x^2 - 4x + 3 + < 0 \left[ \because \left( x - 2 \right) > 0 \text { & }- 2 < 0 \right]\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 3 \right) < 0\]
\[ \Rightarrow 1 < x < 3\]
\[ \Rightarrow x \in \left( 1, 3 \right)\]
\[\text { Also, the domain of f }\left( x \right)is\left( 2, \infty \right).\]
\[ \Rightarrow x \in \left( 1, 3 \right) \cap \left( 2, \infty \right)\]
\[ \Rightarrow x \in \left( 2, 3 \right)\]
APPEARS IN
संबंधित प्रश्न
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
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:
10 − 6x − 2x2
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
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) = x4 − 4x3 + 4x2 + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
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(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing 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
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
Function given by f(x) = sin x is strictly increasing in.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
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)
The function f(x) = sin4x + cos4x is an increasing function if ______.
A function \[f\] is monotonic in an interval when it is which of the following?
As one moves from left to right on a graph, what does a decrease in the \[y\]-values indicate?
