Advertisements
Advertisements
प्रश्न
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Advertisements
उत्तर
f(x) = 2x3 – 3x2 – 12x + 6
∴ f'(x) = `d/dx(2x^3 - 3x^2 - 12x + 6)`
= 2 × 3x2 – 3 × 2x – 12 × 1 + 0
= 6x2 – 6x – 12
= 6(x2 – x – 2)
f is strictly decreasing if f'(x) < 0
i.e. if 6(x2 – x – 2) < 0
i.e. if x2 – x – 2 < 0
i.e. if x2 – x < 2
i.e. if `x^2 - x + (1)/(4) < 2 + (1)/(4)`
i.e. if `(x - 1/2)^2 < (9)/(4)`
i.e. if `-(3)/(2) < x - (1)/(2) < (3)/(2)`
i.e. if `-(3)/(2) + (1)/(2) < x -(1)/(2) + (1)/(2) < (3)/(2) + (1)/(2)`
i.e. if – 1 < x < 2
∴ f is strictly decreasing if – 1 < x < 2
APPEARS IN
संबंधित प्रश्न
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
The interval in which y = x2 e–x is increasing is ______.
Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = x3 − 15x2 + 75x − 50 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 function f given by f(x) = x − [x] is increasing in (0, 1) ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
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 values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
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
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
If f(x) = x3 – 15x2 + 84x – 17, then ______.
y = x(x – 3)2 decreases for the values of x given by : ______.
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The function f (x) = x2, for all real x, is ____________.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
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.
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
The function f(x) = xex(1 − x), x ∈ R, is ______.
Let \[f\] be continuous on \[[a,b]\] and differentiable on \[(a,b)\]. If \[f'(x)\leq0\] for every \[x\in(a,b)\], what follows?
If \[f'(x)<0\] throughout an interval, then \[f\] is
