Giá trị lớn nhất và nhỏ nhất của hàm số: Một số dạng toán và Cách giải

The maximum and minimum values ​​of a function are an extremely important part of the high school math curriculum. So what is the maximum value, the minimum value of the function? How are the maths related to the LWT and the LWT? Let’s Tip.edu.vn Learn about the topic of GTLN and GTNN through the article below!

What is the maximum and minimum value of the function?

Define the maximum and minimum values ​​of the function

Given the function (y=f(x)) defined on the set D

• M is said to be the GTLN of f(x) over D if (left{begin{matrix} f(x)leq M\ exists x_{0}, f(x_{0} = M) end{matrix}right.)
• m is called the GTNN of f(x) on D if (left{begin{matrix} Mleq f(x),, forall x in D\ forall x_{0} in D, f(x_{0}) = m end{matrix}right.)

Method to find the maximum and minimum value of a function

Find GTLN, GTNN of the function y = f(x) defined on the set D

To find GTLN, GTNN of the function y = f(x) on D we calculate y’, find the points where the derivative cancels or does not exist and tabulate the variation. From the variation table, deduce GTLN, GTNN.

Find GTLN and GTNN of a function on a segment

Theorem: Every function that is continuous on a segment has a maximum and a minimum value on that interval

The rule to find GTLN and GTNN of a continuous function f(x) on a segment [a;b]

• Find the points (x_{i} in (a;b), (i=1,2,…,n)) where (f'(x_{i}) = 0) or (f'(x_{i})) is undefined.
• Calculate (f'(x), f(b), f(x_{i}), (i=1,2,…,n))
• Then:
  • (underset{[a;b]}{max}f(x) = maxleft { f(a), f(b),f(x_{i}) right })
  • (underset{[a;b]}{min}f(x) = minleft { f(a), f(b),f(x_{i}) right })

Attention:

• If the function y = f(x) is always increasing or always decreasing over [a;b] then (underset{[a;b]}{max} f(x) = max left { f(a), f(b) right }), (underset{[a;b]}{min} f(x) = min left { f(a), f(b) right }).
• If the function y = f(x) is a periodic function of period T, then to find its GTLN, its value on D, we only need to find its GTLN and GTNN on a segment in D with length equal to T.
• Given the function y = f(x) defined on D. When we put the sub-hidden t = u(x), we find (tin E , forall xin D), we have y = g

Example and how to solve the problem of the maximum and minimum values ​​of the function

Example 1: Find the maximum and minimum value of the function (f(x) = -x^3+4x^2-5x+1) on the segment [1;3]

Solution:

We have (f'(x) = -3x^2+8x-5)

(f'(x) = 0 Leftrightarrow -3x^2 + 8x – 5 = 0 Leftrightarrow x = 1 notin (1;3)) or (x = frac{5}{3} in (1;3))

We have:

(f(1) = -1, f(frac{5}{3}) = -frac{23}{27}, f(3) = -5)

So (underset{[1;3]}{max}f(x) = -frac{23}{27} , when , x=frac{5}{3})

(underset{[1;3]}{min}f(x) =-5 , when , x=3)

Example 2: Find the value of the function (f(x) = frac{4}{3}sin ^3x -sin^2x + frac{2}{3}) on ([0;pi ])

Solution:

Example 3: Find the GTLN and GTNN of the function (f(x) = 2x + sqrt{5-x^2})

Solution:

Determination set (D = [-sqrt{5};sqrt{5}])

We have: (f'(x) = 2-frac{x}{sqrt{5-x^2}}= frac{2sqrt{5-x^2}-x}{sqrt{ 5-x^2}})

(f'(x) = 0 Leftrightarrow 2sqrt{5-x^2} – x =0 Leftrightarrow 2sqrt{5-x^2} = x)

(Leftrightarrow left{begin{matrix} xgeq 0\ 4(5-x^2) = x^2 end{matrix}right Leftrightarrow left{begin{matrix} xgeq 0\ 5x^2-20 =0 end{matrix}right.)

(left{begin{matrix} xgeq 0\ left[begin{array}{l} x=2 \ x=-2 end{array}right. end{matrix}right.)

(Leftrightarrow x=2in (-sqrt{5};sqrt{5}))

Ta có: (f(-sqrt{5}) = -2sqrt{5}; f(2) = 5; f(sqrt{5}) = 2sqrt{5})

Vậy (underset{[-sqrt{5};sqrt{5}]}{max} f(x) = 5, when, x=2)

(underset{[-sqrt{5};sqrt{5}]}{min} f(x) = -2sqrt{5}, when, x=-sqrt{5})

Above are the knowledge related to the topic GTLN and GTNN of functions. Hopefully, it has provided you with useful information for your own learning and research about the largest and smallest GT of the function. Good luck with your studies!

See details through the lecture below by Mr. Nguyen Quoc Chi:

https://www.youtube.com/watch?v=rRu90cZFlTA
(Source: www.youtube.com)

See more:

• What is the monotony of a function? Monotony of quadratic and trigonometric functions
• What is the extremum of the function? Extremes of 3rd and 4th degree functions and Extremes of trigonometric functions