Question

(a)
Find the absolute maximum value and the absolute minimum value of:
1/3x^3- 3 x^2+ 5 x + 8 in [0, 4].

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Answer 1

hey bro..

if u want to find maximum or minimum value of any function.

so u have to differentiate it once.

then find the roots of differencing equation.

now differentiate one more time.

then u got f"(x).

now put the roots of f'(x) in the equation of f"(x).

if by putting the root f"(x) gives u negative value then if u put that root in the giveb function. u will get maximum value of that function.

and if f"(x) gives u positive value then I u put that root in given function.u will get minimum value.....

thanks

Answer 2

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Absolute maxima = 31/3

Absolute minima = 4/3

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step-by-step explanation:

g(x) = ${x}^{3}$/3 - 3${x}^{2}$ + 5x + 8

All we really need to do here is..

✍️ Check whether it is continuous in the given interval ir not.

✍️ So, first notice that this is a polynomial and so is continuous everywhere and therefore is continuous on the given interval.

✍️ Now, we need to get the derivative so that we can find the critical points of the function.

g′(x)=${x}^{2}$ - 6x + 5

=(x -1)(x -5)

now,

It looks like we’ll have two critical points, 

x = 1 and x = 5

✍️ Note that we actually want something more than just the critical points.

✍️ We only want the critical points of the function that lie in the interval in question.

✍️ Both of these do not fall in the interval, so we will use only one of them i.e.,

x = 1

✍️ That may seem like a silly thing to mention at this point, but it is often forgotten, usually when it becomes important, and so we will mention it at every opportunity to make sure it’s not forgotten.

✍️ Now we evaluate the function at the critical points and the end points of the interval.

g(0)= 8

g(1) = 31/3

g(4) = 4/3

✍️ Absolute maxima are the largest and smallest the function will ever be and these three points represent the only places in the interval where the absolute extrema can occur.

✍️ So, from this list we see that the absolute maximum of g(x)

is 31/3 and it occurs at 

x = 1 (a critical point)

and,

the absolute minimum of g(x)

is 4/3 which occurs at 

x = 4 (an endpoint).