Advertisements
Advertisements
प्रश्न
Let f(x) = x3 − 6x2 + 15x + 3. Then,
विकल्प
f(x) > 0 for all x ∈ R
f(x) > f(x + 1) for all x ∈ R
f(x) is invertible
none of these
Advertisements
उत्तर
f(x) is invertible
f(x) =x3 − 6x2 + 15x + 3
\[f'(x) = 3 x^2 - 12x + 15\]
\[ = 3\left( x^2 - 4x + 5 \right)\]
\[ = 3\left( x^2 - 4x + 4 + 1 \right)\]
\[ = 3 \left( x - 2 \right)^2 + \frac{1}{3} > 0\]
\[\text { Therefore, f(x) is strictly increasing function }. \]
\[ \Rightarrow f^{- 1} (x) \text { exists } . \]
\[\text { Hence, f(x) is an invertible function } .\]
APPEARS IN
संबंधित प्रश्न
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
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) = x3 − 6x2 + 9x + 15 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
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 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
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.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
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 ______.
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
For \[f'(x)=12(x-3)(x+2)\], what is the nature of \[f\] on \[(-2,3)\]?
