Advertisements
Advertisements
प्रश्न
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
विकल्प
`[–1, oo)`
[– 2, – 1]
`(-oo, -2]`
[– 1, 1]
Advertisements
उत्तर
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is [– 2, – 1].
Explanation:
The given function is f(x) = 2x3 + 9x2 + 12x – 1
f'(x) = 6x2 + 18x + 12
For increasing and decreasing f'(x) = 0
∴ 6x2 + 18x + 12 = 0
⇒ x2 + 3x + 2 = 0
⇒ x2 + 2x + x + 2 = 0
⇒ x(x + 2) + 1(x + 2) = 0
⇒ (x + 2)(x + 1) = 0
⇒ x = – 2, x = – 1
The possible intervals are `(–oo, – 2), (– 2, – 1), (– 1, oo)`
Now f'(x) = (x + 2) (x + 1)
⇒ `"f'"(x)_((-oo"," -2))` = (–) (–) = (+) increasing
⇒ `"f'"(x)_((-2"," -1))` = (+) (–) = (–) decreasing
⇒ `"f'"(x)_((-1"," oo))` = (+) (+) = (+) increasing.
APPEARS IN
संबंधित प्रश्न
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) = 3x + 17 is strictly increasing on R.
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
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.
Find the intervals in which the function f given by `f(x) = (4sin x - 2x - x cos x)/(2 + cos x)` 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
Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?
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) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
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) ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
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] ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
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
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?
The function f(x) = tan-1 x is ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` 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.
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))` = ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
The function f(x) = xex(1 − x), x ∈ R, is ______.
