Advertisements
Advertisements
प्रश्न
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
विकल्प
(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
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Prove that the logarithmic function is strictly increasing on (0, ∞).
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
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) = x8 + 6x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
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 ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
Let f(x) = x3 − 6x2 + 15x + 3. Then,
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.
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
For every value of x, the function f(x) = `1/7^x` is ______
The function `1/(1 + x^2)` is increasing in the interval ______
Which of the following functions is decreasing on `(0, pi/2)`?
The function f(x) = tanx – x ______.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
Function given by f(x) = sin x is strictly increasing in.
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
The function f(x) = sin4x + cos4x is an increasing function if ______.
