Engineering Maxima And Minima

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MAXIMA AND MINIMA
P r e s e n t e d B y :
K a v e r i H a r i s h B a b u
2 0 G 2 1 A 0 5 7 3

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MaximaandMinimaofFunctionsofTwoIndependentVariables
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• Let f(x,y) be a function of two
independent variables x and y, which
is continuous for all values of x and
y in the neighborhood of (a,b) i.e.
(a+h,b+k) be a point in its
neighborhood which lies inside the
region R.

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• The point (a,b) is called a point of relative
minimum, if f(a,b) ≤ f(a+h,b+k) for all h,k
Then f(a,b) is called the relative minimum value.
• The point (a,b) is called a point of relative
maximum, if f(a,b) ≥ f(a+h,b+k) for all h,k
Then f(a,b) is called the relative minimum value.

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•Stationary point: The point at which function
is either maximum or minimum is known as
stationary point.
•Extreme Value: The value of the function at
stationary point is known as extreme value of
the function.

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Working Rule
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To determine the maxima and minima (extreme values) of a function f (x, y)
Step I : Solve
𝜕𝑓
𝜕𝑥
=0 &
𝜕𝑓
𝜕𝑦
=0 simultaneously for x and y
Step II: Obtain the values of
r=
𝜕2𝑓
𝜕𝑥2, s=
𝜕2𝑓
𝜕𝑦2, t=
𝜕2𝑓
𝜕𝑧2

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.
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•Step III:
1) If 𝒓𝒕 − 𝒔𝟐
> 0 & r < 0 at (a,b) , then f(x,y) is maximum at
(a,b) & maximum value of the function is f (a,b).
2) If 𝒓𝒕 − 𝒔𝟐 > 0 & r > 0 at (a,b) , then f(x,y) is maximum at
(a,b) & maximum value of the function is f (a,b).
3) If 𝒓𝒕 − 𝒔𝟐 < 0 at (a,b) , then f (x,y) is neither maximum nor
minimum at (a,b) .Such point is called Saddle Point.
4) If 𝒓𝒕 − 𝒔𝟐 < 0 at (a,b) , then no conclusion can be made
about the extreme values of f (x,y) & further investigation is
required.

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Example 1
Q.1 Discuss the maxima and Minima of the function 𝑥2 + 𝑦2 + 6𝑥 + 12
Answer: f(x,y)= 𝑥2 + 𝑦2 + 6𝑥 + 12
Step 1: For extreme values
𝝏𝒇
𝝏𝒙
= 0
⇒ 2x+6=0, 2(x+3)=0
x = -3
&
𝝏𝒇
𝝏𝒚
= 0
⇒ 2y=0, y=0

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Therefore stationary point is (-3,0)
STEP II:
r=
𝜕2𝑓
𝜕𝑥2=2,
s=
𝜕2𝑓
𝜕𝑦2=0,
t=
𝜕2𝑓
𝜕𝑧2=2.

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STEP III:
At (-3,0)
𝒓𝒕 − 𝒔𝟐
=2x2-0=4>0
&
r=2>0
Hence f(x,y) is maximum at (-3,0)
f= −3 2
+ 0 2
+6x(-3)+12
=3

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