MATH 121 (Wg Cdr Monir)
Maxima and Minima
♦ Increasing Function:
If y=f(x) is a function of x and y increases as x increases in a certain interval, Then y is called an increasing function of x in that interval.
dydx=tanΨ>0Ψ<90∘
♦ Decreasing Function:
If y=f(x) is a function of x and y decreases as x increases in a certain interval, then y is called a decreasing function of x in that interval.
dydx=tanΨ<0Ψ>90∘
♦ Concavity:
If the graph of a function f lies above all of its tangents on an interval l, then it is said to be concave up on l.
If the graph of a function f lies below all of its tangent on an interval l, then it is said to be concave down.
♦ Test for concavity
• The function f, is concave up on any open interval l where f″(x)>0 and it is concave down where f″(x)<0
• Inflection Point: If x=a inflection point, when
(1) f″(x)>0 when x<a and f″(x)<0 when x>a
or
(2) f″(x)<0 when x<a and f″(x)>0 when x>a.
Note: Inflection point does not occur where f″(x)=0. The inflection point occurs where the sign of the second derivative change.
Question: Find the maximum and minimum value of the function x5−5x4+5x3−1. Also, find the inflection point if any.
Solution: Let,
f(x)=x5−5x4+5x3−1f′(x)=5x4−20x3+15x2
For maximum or minimum f′(x)=0
5x4−20x3+15x2=0or, 5x2(x2−4x+3)=0or, 5x2(x−1)(x−3)=0or, x=0,1,3
Now, f″(x)=20x3−60x2+30x
For x = 1
f″(x)=20−60+30=−10<0
then the function f(x) is maximum at x = 1 and the maximum value is, f(x)=15−5×14+5×13−1=0
For x = 3
f″(x)=20×33−60×33+30×3=90>0
then the function f(x) is minimum at x = 3 and the minimum value is,
f(x)=x5−5×34+5×33−1=−28
For x = 0
f″(x)=20×0−60×0+30×0=0f″(x)=20x3−60xpe2+30xf‴(x)=60x3−120x+30Now for x=0,f‴(x)=30≠0
the function has neither maximum nor minimum at x = 0
At x = 0 is an inflection point
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