Advertisements
Advertisements
Question
Find the maximum and minimum values of f(x) = secx + log cos2x, 0 < x < 2π
Advertisements
Solution
f(x) = secx + 2 log cosx
Therefore, f'(x) = secx tanx – 2 tanx = tanx (secx –2)
f'(x) = 0
⇒ tanx = 0 or secx = 2 or cosx = `1/2`
Therefore, possible values of x are x = 0
or x = π and x = `pi/3` or x = `(5pi)/3`
Again, f′(x) = sec2x (secx –2) + tanx (secx tanx)
= sec3x + secx tan2x – 2sec2x
= secx (sec2x + tan2x – 2secx).
We note that
f′(0) = 1(1 + 0 – 2) = –1 < 0. Therefore, x = 0 is a point of maxima.
f′(π) = –1(1 + 0 + 2) = –3 < 0. Therefore, x = π is a point of maxima.
`"f'"(pi/3)` = 2(4 + 3 – 4) = 6 > 0. Therefore, x = `pi/3` is a point of minima.
`"f'"((5pi)/3)` = 2(4 + 3 – 4) = 6 > 0. Therefore, x = `(5pi)/3` is a point of minima.
Maximum Value of y at x = 0 is 1 + 0 = 1
Maximum Value of y at x = π is –1 + 0 = –1
Minimum Value of y at x = `pi/3` is `2 + 2 log 1/2` = 2(1 – log2)
Minimum Value of y at x = `(5pi)/3` is `2 + 2 log 1/2` = 2(1 – log2)
APPEARS IN
RELATED QUESTIONS
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 cylinder is `4/27 pih^3` tan2α.
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 (x) = 2x2 − 5x + 3 on [1, 3] 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) = x2 − 8x + 12 on [2, 6] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = (x2 − 1) (x − 2) on [−1, 2] ?
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 \[f\left( x \right) = \frac{x}{2} - \sin\frac{\pi x}{6} \text { on }[ - 1, 0]\]?
Using Rolle's theorem, find points on the curve y = 16 − x2, x ∈ [−1, 1], where tangent is parallel to x-axis.
At what point on the following curve, is the tangent parallel to x-axis y = x2 on [−2, 2]
?
At what point on the following curve, is the tangent parallel to x-axis y = \[e^{1 - x^2}\] on [−1, 1] ?
At what point on the following curve, is the tangent parallel to x-axis y = 12 (x + 1) (x − 2) on [−1, 2] ?
If f : [−5, 5] → R is differentiable and if f' (x) doesnot vanish anywhere, then prove that f (−5) ± f (5) ?
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) = x3 − 2x2 − x + 3 on [0, 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 theorem f(x) = x(x −1) on [1, 2] ?
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(x) = (x − 1)(x − 2)(x − 3) on [0, 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 theore f(x) = tan−1 x on [0, 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 theorem f(x) = sin x − sin 2x − x on [0, π] ?
Find a point on the parabola y = (x − 4)2, where the tangent is parallel to the chord joining (4, 0) and (5, 1) ?
If f (x) = Ax2 + Bx + C is such that f (a) = f (b), then write the value of c in Rolle's theorem ?
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 ______.
The value of c in Lagrange's mean value theorem for the function f (x) = x (x − 2) when x ∈ [1, 2] is
A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of types A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 4 hours available for assembling. The profit is ₹ 50 each for type A and ₹60 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize profit? Formulate the above LPP and solve it graphically and find the maximum profit.
The values of a for which y = x2 + ax + 25 touches the axis of x are ______.
The least value of the function f(x) = `"a"x + "b"/x` (where a > 0, b > 0, x > 0) is ______.
The minimum value of `1/x log x` in the interval `[2, oo]` is
Let y = `f(x)` be the equation of a curve. Then the equation of tangent at (xo, yo) is :-
For a continuous function on the closed interval \[a,d\], what does \[f(d)\] represent?
Which statement about local extrema and absolute extrema is correct?
Which statement expresses the Extreme Value Theorem?
Which points are critical points for a function \[f\]?
What is \[f(1)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
