Advertisements
Advertisements
Question
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Advertisements
Solution
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
RELATED QUESTIONS
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
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(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) = x3 − 6x2 + 9x + 15 ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
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 < \[\frac{\pi}{2}\] ?
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 a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
The slope of tangent at any point (a, b) is also called as ______.
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
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 ______.
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Which of the following functions is decreasing on `(0, pi/2)`?
The function f(x) = tan-1 x is ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` 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.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
As one moves from left to right on a graph, what do constant \[y\]-values indicate?
