Question

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
f (x) = (x − 1)² + 3, x ∈ [−3, 1]

Answer 1

Answer:

$\huge\underline{\underline{\mathbb\pink{ANSWER}}}$

ANSWER

f′(x)=−2cosxsinx+cosx

Putting this to zero, we get

f′(x)=−2cosxsinx+cosx

⇒−2cosxsinx+cosx=0

⇒cosx(2sinx−1)

⇒x=6π or 2π

Now let's evaluate the value of the function at critical points and at extreme points of domain.

f(6π)=cos2(6π)+sin(6π)=45

f(2π)=cos2(2π)+sin(2π)=1

f(0)=cos2(0)+sin(0)=1

f(π)=cos2(π)+sin(π)=1

And we can see that Function will have maxima at x=6π and will have minima at x=2π,0,π

Answer 2

dy/dx=2(x-1)=0
x=1/2
So minimum value is when x=1/2
i.e 13/4