Calculus Examples

Find the Absolute Max and Min over the Interval y=4 cube root of x-1-3 ; [-10,10]
y=43x-1-3y=43x13 ; [-10,10][10,10]
Step 1
Find the critical points.
Tap for more steps...
Step 1.1
Find the first derivative.
Tap for more steps...
Step 1.1.1
Find the first derivative.
Tap for more steps...
Step 1.1.1.1
By the Sum Rule, the derivative of 43x-1-343x13 with respect to xx is ddx[43x-1]+ddx[-3]ddx[43x1]+ddx[3].
ddx[43x-1]+ddx[-3]ddx[43x1]+ddx[3]
Step 1.1.1.2
Evaluate ddx[43x-1]ddx[43x1].
Tap for more steps...
Step 1.1.1.2.1
Use nax=axnnax=axn to rewrite 3x-13x1 as (x-1)13(x1)13.
ddx[4(x-1)13]+ddx[-3]ddx[4(x1)13]+ddx[3]
Step 1.1.1.2.2
Since 44 is constant with respect to xx, the derivative of 4(x-1)134(x1)13 with respect to xx is 4ddx[(x-1)13]4ddx[(x1)13].
4ddx[(x-1)13]+ddx[-3]4ddx[(x1)13]+ddx[3]
Step 1.1.1.2.3
Differentiate using the chain rule, which states that ddx[f(g(x))]ddx[f(g(x))] is f(g(x))g(x)f'(g(x))g'(x) where f(x)=x13f(x)=x13 and g(x)=x-1g(x)=x1.
Tap for more steps...
Step 1.1.1.2.3.1
To apply the Chain Rule, set uu as x-1x1.
4(ddu[u13]ddx[x-1])+ddx[-3]4(ddu[u13]ddx[x1])+ddx[3]
Step 1.1.1.2.3.2
Differentiate using the Power Rule which states that ddu[un]ddu[un] is nun-1nun1 where n=13n=13.
4(13u13-1ddx[x-1])+ddx[-3]4(13u131ddx[x1])+ddx[3]
Step 1.1.1.2.3.3
Replace all occurrences of uu with x-1x1.
4(13(x-1)13-1ddx[x-1])+ddx[-3]4(13(x1)131ddx[x1])+ddx[3]
4(13(x-1)13-1ddx[x-1])+ddx[-3]4(13(x1)131ddx[x1])+ddx[3]
Step 1.1.1.2.4
By the Sum Rule, the derivative of x-1x1 with respect to xx is ddx[x]+ddx[-1]ddx[x]+ddx[1].
4(13(x-1)13-1(ddx[x]+ddx[-1]))+ddx[-3]4(13(x1)131(ddx[x]+ddx[1]))+ddx[3]
Step 1.1.1.2.5
Differentiate using the Power Rule which states that ddx[xn]ddx[xn] is nxn-1nxn1 where n=1n=1.
4(13(x-1)13-1(1+ddx[-1]))+ddx[-3]4(13(x1)131(1+ddx[1]))+ddx[3]
Step 1.1.1.2.6
Since -11 is constant with respect to xx, the derivative of -11 with respect to xx is 00.
4(13(x-1)13-1(1+0))+ddx[-3]4(13(x1)131(1+0))+ddx[3]
Step 1.1.1.2.7
To write -11 as a fraction with a common denominator, multiply by 3333.
4(13(x-1)13-133(1+0))+ddx[-3]4(13(x1)13133(1+0))+ddx[3]
Step 1.1.1.2.8
Combine -11 and 3333.
4(13(x-1)13+-133(1+0))+ddx[-3]4(13(x1)13+133(1+0))+ddx[3]
Step 1.1.1.2.9
Combine the numerators over the common denominator.
4(13(x-1)1-133(1+0))+ddx[-3]4(13(x1)1133(1+0))+ddx[3]
Step 1.1.1.2.10
Simplify the numerator.
Tap for more steps...
Step 1.1.1.2.10.1
Multiply -11 by 33.
4(13(x-1)1-33(1+0))+ddx[-3]4(13(x1)133(1+0))+ddx[3]
Step 1.1.1.2.10.2
Subtract 33 from 11.
4(13(x-1)-23(1+0))+ddx[-3]4(13(x1)23(1+0))+ddx[3]
4(13(x-1)-23(1+0))+ddx[-3]4(13(x1)23(1+0))+ddx[3]
Step 1.1.1.2.11
Move the negative in front of the fraction.
4(13(x-1)-23(1+0))+ddx[-3]4(13(x1)23(1+0))+ddx[3]
Step 1.1.1.2.12
Add 11 and 00.
4(13(x-1)-231)+ddx[-3]4(13(x1)231)+ddx[3]
Step 1.1.1.2.13
Combine 1313 and (x-1)-23(x1)23.
4((x-1)-2331)+ddx[-3]4((x1)2331)+ddx[3]
Step 1.1.1.2.14
Multiply (x-1)-233(x1)233 by 11.
4(x-1)-233+ddx[-3]4(x1)233+ddx[3]
Step 1.1.1.2.15
Move (x-1)-23(x1)23 to the denominator using the negative exponent rule b-n=1bnbn=1bn.
413(x-1)23+ddx[-3]413(x1)23+ddx[3]
Step 1.1.1.2.16
Combine 44 and 13(x-1)2313(x1)23.
43(x-1)23+ddx[-3]43(x1)23+ddx[3]
43(x-1)23+ddx[-3]43(x1)23+ddx[3]
Step 1.1.1.3
Differentiate using the Constant Rule.
Tap for more steps...
Step 1.1.1.3.1
Since -33 is constant with respect to xx, the derivative of -33 with respect to xx is 00.
43(x-1)23+043(x1)23+0
Step 1.1.1.3.2
Add 43(x-1)2343(x1)23 and 00.
f(x)=43(x-1)23f'(x)=43(x1)23
f(x)=43(x-1)23f'(x)=43(x1)23
f(x)=43(x-1)23f'(x)=43(x1)23
Step 1.1.2
The first derivative of f(x)f(x) with respect to xx is 43(x-1)2343(x1)23.
43(x-1)2343(x1)23
43(x-1)2343(x1)23
Step 1.2
Set the first derivative equal to 00 then solve the equation 43(x-1)23=043(x1)23=0.
Tap for more steps...
Step 1.2.1
Set the first derivative equal to 00.
43(x-1)23=043(x1)23=0
Step 1.2.2
Set the numerator equal to zero.
4=04=0
Step 1.2.3
Since 4040, there are no solutions.
No solution
No solution
Step 1.3
Find the values where the derivative is undefined.
Tap for more steps...
Step 1.3.1
Apply the rule xmn=nxmxmn=nxm to rewrite the exponentiation as a radical.
433(x-1)2433(x1)2
Step 1.3.2
Set the denominator in 433(x-1)2433(x1)2 equal to 00 to find where the expression is undefined.
33(x-1)2=033(x1)2=0
Step 1.3.3
Solve for xx.
Tap for more steps...
Step 1.3.3.1
To remove the radical on the left side of the equation, cube both sides of the equation.
(33(x-1)2)3=03(33(x1)2)3=03
Step 1.3.3.2
Simplify each side of the equation.
Tap for more steps...
Step 1.3.3.2.1
Use nax=axnnax=axn to rewrite 3(x-1)23(x1)2 as (x-1)23(x1)23.
(3(x-1)23)3=03(3(x1)23)3=03
Step 1.3.3.2.2
Simplify the left side.
Tap for more steps...
Step 1.3.3.2.2.1
Simplify (3(x-1)23)3(3(x1)23)3.
Tap for more steps...
Step 1.3.3.2.2.1.1
Apply the product rule to 3(x-1)233(x1)23.
33((x-1)23)3=0333((x1)23)3=03
Step 1.3.3.2.2.1.2
Raise 33 to the power of 33.
27((x-1)23)3=0327((x1)23)3=03
Step 1.3.3.2.2.1.3
Multiply the exponents in ((x-1)23)3((x1)23)3.
Tap for more steps...
Step 1.3.3.2.2.1.3.1
Apply the power rule and multiply exponents, (am)n=amn(am)n=amn.
27(x-1)233=0327(x1)233=03
Step 1.3.3.2.2.1.3.2
Cancel the common factor of 33.
Tap for more steps...
Step 1.3.3.2.2.1.3.2.1
Cancel the common factor.
27(x-1)233=03
Step 1.3.3.2.2.1.3.2.2
Rewrite the expression.
27(x-1)2=03
27(x-1)2=03
27(x-1)2=03
27(x-1)2=03
27(x-1)2=03
Step 1.3.3.2.3
Simplify the right side.
Tap for more steps...
Step 1.3.3.2.3.1
Raising 0 to any positive power yields 0.
27(x-1)2=0
27(x-1)2=0
27(x-1)2=0
Step 1.3.3.3
Solve for x.
Tap for more steps...
Step 1.3.3.3.1
Divide each term in 27(x-1)2=0 by 27 and simplify.
Tap for more steps...
Step 1.3.3.3.1.1
Divide each term in 27(x-1)2=0 by 27.
27(x-1)227=027
Step 1.3.3.3.1.2
Simplify the left side.
Tap for more steps...
Step 1.3.3.3.1.2.1
Cancel the common factor of 27.
Tap for more steps...
Step 1.3.3.3.1.2.1.1
Cancel the common factor.
27(x-1)227=027
Step 1.3.3.3.1.2.1.2
Divide (x-1)2 by 1.
(x-1)2=027
(x-1)2=027
(x-1)2=027
Step 1.3.3.3.1.3
Simplify the right side.
Tap for more steps...
Step 1.3.3.3.1.3.1
Divide 0 by 27.
(x-1)2=0
(x-1)2=0
(x-1)2=0
Step 1.3.3.3.2
Set the x-1 equal to 0.
x-1=0
Step 1.3.3.3.3
Add 1 to both sides of the equation.
x=1
x=1
x=1
x=1
Step 1.4
Evaluate 43x-1-3 at each x value where the derivative is 0 or undefined.
Tap for more steps...
Step 1.4.1
Evaluate at x=1.
Tap for more steps...
Step 1.4.1.1
Substitute 1 for x.
43(1)-1-3
Step 1.4.1.2
Simplify.
Tap for more steps...
Step 1.4.1.2.1
Simplify each term.
Tap for more steps...
Step 1.4.1.2.1.1
Subtract 1 from 1.
430-3
Step 1.4.1.2.1.2
Rewrite 0 as 03.
4303-3
Step 1.4.1.2.1.3
Pull terms out from under the radical, assuming real numbers.
40-3
Step 1.4.1.2.1.4
Multiply 4 by 0.
0-3
0-3
Step 1.4.1.2.2
Subtract 3 from 0.
-3
-3
-3
Step 1.4.2
List all of the points.
(1,-3)
(1,-3)
(1,-3)
Step 2
Evaluate at the included endpoints.
Tap for more steps...
Step 2.1
Evaluate at x=-10.
Tap for more steps...
Step 2.1.1
Substitute -10 for x.
43(-10)-1-3
Step 2.1.2
Simplify each term.
Tap for more steps...
Step 2.1.2.1
Subtract 1 from -10.
43-11-3
Step 2.1.2.2
Rewrite -11 as (-1)311.
Tap for more steps...
Step 2.1.2.2.1
Rewrite -11 as -1(11).
43-1(11)-3
Step 2.1.2.2.2
Rewrite -1 as (-1)3.
43(-1)311-3
43(-1)311-3
Step 2.1.2.3
Pull terms out from under the radical.
4(-1311)-3
Step 2.1.2.4
Rewrite -1311 as -311.
4(-311)-3
Step 2.1.2.5
Multiply -1 by 4.
-4311-3
-4311-3
-4311-3
Step 2.2
Evaluate at x=10.
Tap for more steps...
Step 2.2.1
Substitute 10 for x.
43(10)-1-3
Step 2.2.2
Subtract 1 from 10.
439-3
439-3
Step 2.3
List all of the points.
(-10,-4311-3),(10,439-3)
(-10,-4311-3),(10,439-3)
Step 3
Compare the f(x) values found for each value of x in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest f(x) value and the minimum will occur at the lowest f(x) value.
Absolute Maximum: (10,439-3)
Absolute Minimum: (-10,-4311-3)
Step 4
image of graph
;
(
(
)
)
|
|
[
[
]
]
7
7
8
8
9
9
°
°
θ
θ
4
4
5
5
6
6
/
/
^
^
×
×
>
>
π
π
1
1
2
2
3
3
-
-
+
+
÷
÷
<
<
!
!
,
,
0
0
.
.
%
%
=
=
 [x2  12  π  xdx ]