Advertisements
Advertisements
प्रश्न
Find the absolute maximum and absolute minimum values of the function f given by f(x)=sin2x-cosx,x ∈ (0,π)
Advertisements
उत्तर
f(x)=sin2x-cosx
f'(x)=2 sinx.cosx+sinx
=sinx(2cosx+1)
Equating f’(x) to zero.
f'(x)=0
sin x(2cos x + 1) = 0
sin x = 0
∴ x = 0, π
`2cos x + 1 = 0`
`⇒cos x =-1/2`
`therefore x=(5pi)/6`
`f(0) = sin20 – cos 0 = − 1`
`f((5pi)/6)=sin^2(5pi/6)-cos((5pi)/6)`
`=sin^2(pi/6)+cos(pi/6)`
`=1/4-sqrt3/2`
`=((1-2sqrt3)/sqrt4)`
`f(pi) = sin^2pi – cospi = 1`
Of these values, the maximum value is 1, and the minimum value is −1.
Thus, the absolute maximum and absolute minimum values of f(x) are 1 and −1, which it attains at x = 0 and x = π.
APPEARS IN
संबंधित प्रश्न
f (x) = sin \[\frac{1}{x}\] for −1 ≤ x ≤ 1 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(x − 1)2 on [0, 1] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = \[\frac{\sin x}{e^x}\] on 0 ≤ x ≤ π ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = \[{e^{1 - x}}^2\] on [−1, 1] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x − sin 2x on [0, π]?
At what point on the following curve, is the tangent parallel to x-axis y = \[e^{1 - x^2}\] on [−1, 1] ?
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 theore \[f\left( x \right) = \sqrt{25 - x^2}\] on [−3, 4] ?
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 \[f\left( x \right) = x + \frac{1}{x} \text { on }[1, 3]\] ?
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 \[f\left( x \right) = \sqrt{x^2 - 4} \text { on }[2, 4]\] ?
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) = x3 − 5x2 − 3x on [1, 3] ?
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) ?
Find a point on the curve y = x3 + 1 where the tangent is parallel to the chord joining (1, 2) and (3, 28) ?
Using Lagrange's mean value theorem, prove that (b − a) sec2 a < tan b − tan a < (b − a) sec2 b
where 0 < a < b < \[\frac{\pi}{2}\] ?
If the polynomial equation \[a_0 x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + . . . + a_2 x^2 + a_1 x + a_0 = 0\] n positive integer, has two different real roots α and β, then between α and β, the equation \[n \ a_n x^{n - 1} + \left( n - 1 \right) a_{n - 1} x^{n - 2} + . . . + a_1 = 0 \text { has }\].
The value of c in Rolle's theorem when
f (x) = 2x3 − 5x2 − 4x + 3, x ∈ [1/3, 3] is
Show that the local maximum value of `x + 1/x` is less than local minimum value.
Find the area of greatest rectangle that can be inscribed in an ellipse `x^2/"a"^2 + y^2/"b"^2` = 1
An isosceles triangle of vertical angle 2θ is inscribed in a circle of radius a. Show that the area of triangle is maximum when θ = `pi/6`
The values of a for which y = x2 + ax + 25 touches the axis of x are ______.
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`
If the graph of a differentiable function y = f (x) meets the lines y = – 1 and y = 1, then the graph ____________.
For a continuous function on the closed interval \[a,d\], what does \[f(d)\] represent?
Which statement expresses the Extreme Value Theorem?
For \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\], which critical point is obtained from \[2(8x-1)=0\]?
What is \[f\left(\frac{1}{8}\right)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
What is \[f(0)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
What are the absolute extrema of \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\] on \[-1,1\]?
