Advertisements
Advertisements
Question
Find the difference between the greatest and least values of the function f(x) = sin2x – x, on `[- pi/2, pi/2]`
Advertisements
Solution
f(x) = sin2x – x
⇒ f′(x) = 2 cos2x – 1
Therefore, f′(x) = 0
⇒ cos2x = `1/2`
⇒ 2x is `(-pi)/3` or `pi/3`
⇒ x = `- pi/6` or `pi/6`
`"f"(- pi/2) = sin(- pi) + pi/2 = pi/2`
`"f"(- pi/6) = sin(-(2pi)/6) + pi/6 - - sqrt(3)/2 + pi/6`
`"f"(pi/6) = sin((2pi)/6) - pi/6 = sqrt(3)/2 - pi/6`
`"f"(pi/2) = sin(pi) - pi/2 = - pi/2`
Clearly, `pi/2` is the greatest value and `- pi/2` is the least.
Therefore, difference = `pi/2 + pi/2` = π
APPEARS IN
RELATED QUESTIONS
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of ______.
f (x) = [x] for −1 ≤ x ≤ 1, where [x] denotes the greatest integer not exceeding x Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
\[f\left( x \right) = \begin{cases}- 4x + 5, & 0 \leq x \leq 1 \\ 2x - 3, & 1 < x \leq 2\end{cases}\] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = (x − 1) (x − 2)2 on [1, 2] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = x(x − 1)2 on [0, 1] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = x(x − 4)2 on the interval [0, 4] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = x(x −2)2 on the interval [0, 2] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = x2 + 5x + 6 on the interval [−3, −2] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin 3x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x + cos x on [0, π/2] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = x2 − 5x + 4 on [1, 4] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x − sin 2x on [0, π]?
It is given that the Rolle's theorem holds for the function f(x) = x3 + bx2 + cx, x \[\in\] at the point x = \[\frac{4}{3}\] , Find the values of b and c ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = 2x − x2 on [0, 1] ?
Find a point on the parabola y = (x − 4)2, where the tangent is parallel to the chord joining (4, 0) and (5, 1) ?
Find a point on the curve y = x2 + x, where the tangent is parallel to the chord joining (0, 0) and (1, 2) ?
Find the points on the curve y = x3 − 3x, where the tangent to the curve is parallel to the chord joining (1, −2) and (2, 2) ?
If f (x) = Ax2 + Bx + C is such that f (a) = f (b), then write the value of c in Rolle's theorem ?
If 4a + 2b + c = 0, then the equation 3ax2 + 2bx + c = 0 has at least one real root lying in the interval
The value of c in Rolle's theorem when
f (x) = 2x3 − 5x2 − 4x + 3, x ∈ [1/3, 3] is
When the tangent to the curve y = x log x is parallel to the chord joining the points (1, 0) and (e, e), the value of x is ______.
Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?
Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi-vertical angle α is one-third that of the cone and the greatest volume of the cylinder is `(4)/(27) pi"h"^3 tan^2 α`.
Find the maximum and minimum values of f(x) = secx + log cos2x, 0 < x < 2π
Minimum value of f if f(x) = sinx in `[(-pi)/2, pi/2]` is ______.
Prove that f(x) = sinx + `sqrt(3)` cosx has maximum value at x = `pi/6`
At x = `(5pi)/6`, f(x) = 2 sin3x + 3 cos3x is ______.
It is given that at x = 1, the function x4 - 62x2 + ax + 9 attains its maximum value on the interval [0, 2]. Find the value of a.
The minimum value of `1/x log x` in the interval `[2, oo]` is
Which statement about local extrema and absolute extrema is correct?
If a differentiable function has an absolute max or min at an interior point \[c\], what must be true?
After finding critical points and identifying endpoints, what should be done next?
