Advertisements
Advertisements
प्रश्न
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Advertisements
उत्तर
f(x) = 6 - 9x - x2
f'(x) = - 9x - 2x = -(2x + 9)
f'(x) = 0 ⇒ (2x + 9) = 0 ⇒ x = - `9/2`
The point x = `- 9/2` divides the number line into two parts, intervals `(- oo, - 9/2)` and `(- 9/2, oo)`.
In the interval `(- oo, - 9/2)`, f'(x) = (-)(-) = + Positive
Hence, the function f is continuously increasing.
In the interval `(- 9/2, oo)`, f'(x) = (-)(+) = - Negative
Hence, the function f is continuously decreasing.
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
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 the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
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 ?
Show that f(x) = e2x is increasing on R.
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
Every invertible function is
Function f(x) = ax is increasing on R, if
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
The function f(x) = x9 + 3x7 + 64 is increasing on
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
If f(x) = x3 – 15x2 + 84x – 17, then ______.
The function f(x) = tanx – x ______.
The function f (x) = 2 – 3 x is ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
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).
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
The function f(x) = x3 + 3x is increasing in interval ______.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
