Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i)f(x)=x3,x∈[-2,2] (ii) f(x)=sin x+cos x,x∈[0,π] (iii) f(x)=4x-1/2x2,x∈[-2,\(\frac{9}{2}\)] (iv) f(x)=(x-1)2+3,x∈[-3,1]
(i) The given function is f(x) = x3. f'(x)=3x2 Now, f'(x)=0=x=0 Then, we evaluate the value of f at critical point x = 0 and at end points of the interval [−2, 2]. f(0) = 0 f(−2) = (−2) 3 = −8 f(2) = (2)3 = 8 Hence, we can conclude that the absolute maximum value of f on [−2, 2] is 8 occurring at x = 2. Also, the absolute minimum value of f on [−2, 2] is −8 occurring at x = −2. (ii) The given function is f(x) = sin x + cos x. f'(x)=0=sinx=cos x= tanx=1 x=π/4
Then, we evaluate the value of f at critical point x=π/4 and at the endpoints of the interval [0, π].
f(\(\frac{\pi}{4}\))=sin \(\frac{\pi}{4}\)+cos \(\frac{\pi}{4}\)=\(\frac{1}{\sqrt2}\)+\(\frac{1}{\sqrt2}\)=\(\frac{2}{\sqrt2}\)=√2
f(0)=sin0+cos0+1=1
f(π)=sinπ+cosπ=0-1=-1
Hence, we can conclude that the absolute maximum value of f on [0, π] is occurring at x=\(\frac{\pi}{4}\)
and the absolute minimum value of f on [0, π] is −1 occurring at x = π.
(iii) The given function is f'(x)=0=x=4
Then, we evaluate the value of f at critical point x = 4 and at the endpoints of the interval [-2,\(\frac{9}{2}\)]
Hence, we can conclude that the absolute maximum value of f on [-2,\(\frac{9}{2}\)] is -10 occurring at x=-2
(iv) The given function is f(x)=(x-1)2+3
f(x)=2(x-1)
Now,
f'(x)=0=2(x − 1) = 0 ∴ x = 1
Then, we evaluate the value of f at critical point x = 1 and at the endpoints of the interval [−3, 1].
f(1)=(1-1)2+3=0+3=3
f(-3)=(-3-1)2+3=16+3=19
Hence, we can conclude that the absolute maximum value of f on [−3, 1] is 19 occurring at x = −3 and the minimum value of f on [−3, 1] is 3 occurring at x = 1.
If \( x = a(0 - \sin \theta) \), \( y = a(1 + \cos \theta) \), find \[ \frac{dy}{dx}. \]
Find the least value of ‘a’ for which the function \( f(x) = x^2 + ax + 1 \) is increasing on the interval \( [1, 2] \).
Read the following text carefully:
Union Food and Consumer Affairs Minister said that the Central Government has taken many proactive steps in the past few years to control retail prices of food items. He said that the government aims to keep inflation under control without compromising the country’s economic growth. Retail inflation inched up to a three-month high of 5.55% in November 2023 driven by higher food prices. Inflation has been declining since August 2023, when it touched 6.83%. 140 new price monitoring centres had been set up by the Central Government to keep a close watch on wholesale and retail prices of essential commodities. The Government has banned the export of many food items like wheat, broken rice, non-basmati white rice, onions etc. It has also reduced import duties on edible oils and pulses to boost domestic supply and control price rise. On the basis of the given text and common understanding,
answer the following questions:
The extrema of a function are very well known as Maxima and minima. Maxima is the maximum and minima is the minimum value of a function within the given set of ranges.
There are two types of maxima and minima that exist in a function, such as: