Advertisements
Advertisements
Question
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
Options
(1, 2)
(2, 3)
(1, 3)
(2, 4)
Advertisements
Solution
(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
RELATED QUESTIONS
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
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) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
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(x) = x3 − 6x2 + 9x + 15 ?
Show that f(x) = e2x is increasing on R.
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Let f defined on [0, 1] be twice differentiable such that | f (x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Function f(x) = ax is increasing on R, if
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
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 ______.
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
Which of the following functions is decreasing on `(0, pi/2)`?
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] decreasing at \[x_0\]?
In the procedure for finding intervals, which conclusion is correct when \[f'(x)<0\]?
