Advertisements
Advertisements
प्रश्न
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
पर्याय
increases on [0, a]
decreases on [0, a]
increases on [−a, 0]
decreases on [a, 2a]
Advertisements
उत्तर
Given: ϕ(x) = f(x) + f(2a − x)
Differentiating above equation with respect to x we get,
ϕ'(x) = f'(x) − f(2a − x) .....(1)
Since, f''(x) > 0, f'(x) is an increasing function.
Now,
when \[x \in \left[ 0, a \right]\]
\[f'\left( x \right) \leq f\left( 2a - x \right) . . . . . \left( 2 \right)\]
Considering equation (1) and (2) we get,
ϕ'(x) ≤ 0
⇒ ϕ'(x) is decreasing in [0, a]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
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 intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Function f(x) = ax is increasing on R, if
Function f(x) = loga x is increasing on R, if
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.
Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q
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).
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
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.
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
The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing is ______
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 function f(x) = 9 - x5 - x7 is decreasing for
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
The function f(x) = tanx – x ______.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
Function given by f(x) = sin x is strictly increasing in.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
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 ______.
If f(x) = x + cosx – a then ______.
