Chapter 1.

1.1 Problem Statement

{2,3,4}
{5,6,7,8,9}
$a*$b
1/exp(1)
round(pow(($b*$1e),($a*$1e)),6)

Find the minimum value of f(x) = $b·x$a·x for x > 0.

1.2 Step 1

In order to find any minimum values of f(x), we first need to find the critical points of f(x).

Question Sequence

Question 1.1

A number c in the domain of f is called a critical point if f'(x) is JxoK1BqeuwH1wnvtnpN3OpZ4I/0IrwLIwbagR6VX/B12v51fMTNATsa7ojq9jvde.

Correct.
Incorrect.

Question 1.2

Since the variable x is in the base and in the exponent of the given function, the derivative of f(x) will require nIjF7RFXxoFl8VffuQlv1+l7ExRQs8rWh7C6zGXbMGn346TNiPy16vAKp0E= differentiation.

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.3

Find f'(x).

f(x) = $b·x$a·x

f'(x) = hlF2UPykPxE=·x$a·x(1 + ln(x)))

Correct.
Incorrect.

Question 1.4

Solve f'(x) = 0 to find any critical points, c > 0.

f'(x) = $ab·x$a·x(1 + ln(x)))

c =uAtmqcFxPN+0/SME.

Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.5

When x > 0, $b·x$a·x is AcRpPU+UEYM1XQgxa+Iehy0gqa8= zero and $b·x$a·x is iDztFdIv1inSjCiJ7SFTspbK7I3khNbwYZTJw3C0/RfRBLaqPBvs+VhU7FDOqojo.

Correct.
Incorrect.

Question 1.6

The First Derivative Test for critical points states that for any critical point x = c:

If f'(x) changes sign from + to − at x = c, then f(c) is a local MLrjsAZLmRw2xfA/t8djnqWlqT8=.

If f'(x) changes sign from − to + at x = c, then f(c) is a local cCmYpIf3+uu6kk3FVOz/JXh4+jw=.

Correct.
Incorrect.

1.5 Step 4

Use the critical point, , to divide the real line for x > 0 into two intervals.

,

To determine if yields a maximum or minimum, we need to find the sign of f'(x) in each interval and then use the first derivative test.

Question Sequence

Question 1.7

Since , let's pick x1 = 0.1 in the first interval and x2 = 1 in the second interval. Fill in the table below with the appropriate sign.

Interval x-value Sign of f'(x)
0.1 097XeLvBC5c=
1 brSjF5lOKMQ=
Correct.
Incorrect.

Question 1.8

At , f'(x) changes sign from IdoEMfAMQlPRKpxgEQrxzyjEANn/EfUFN3snk7XRbcBB3VBm9RJtRxh6FbBvk1xg.

Correct.
Incorrect.

1.6 Step 5

Question Sequence

Question 1.9

Thus, the local CoKdRWMwar7l2TXZp/HL1V0+ugY= value of the function is as follows.

Correct.
Incorrect.

Question 1.10

JRTGa0xCnPWOxQpY

(Round your answer to six decimal places.)

Correct.
Incorrect.