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asked May 6, 2015 in CALCULUS by anonymous

5 Answers

0 votes

(1)

Step 1:

Critical number :

A critical number of a function is a number in the domain of such that either or does not exist.

The function is image.

is continuous and differentiable at all values of  because it is a polynomial.

Solutions of are the critical numbers.

Differentiate on each side with respect to .

image

image.

Step 2:

image.

Equate to zero.

image

image

image

image

image.  

Critical number is  image.

Solution:

Critical number is  image.

answered May 6, 2015 by Sammi Mentor
0 votes

(2)

Step 1:

Critical number :

A critical number of a function is a number in the domain of such that either or does not exist.

The function is image.

The domain of a function is all values of image, those makes the function mathematically correct.

There should not be any negative number in the square root.

image

image

image.

The domain of the function is image.

Step 2:

Differentiate image on each side with respect to .

image

Apply product rule in derivatives: image.

image

image.

answered May 6, 2015 by Sammi Mentor
edited May 6, 2015 by Sammi

Contd...

Step 3:

image.

Equate image to zero.

image

image

image

image.

image is not defined at image.

image is in the domain of image.

The critical points are image and image.

Solution:

The critical points are image and image.

0 votes

(3)

Step 1:

The function is image, on the interval image.

Evaluate the critical points.

The function is image.

Differentiate on each side with respect to .

image

image.

Find the critical points, by equate to zero.

image

image

image

image

The critical point is image.

Step 2:

Absolute extrema of a function exist either at the end points or at the critical points.

Substitute the critical point in the function.

Substitute image in .

image

image.

answered May 6, 2015 by Sammi Mentor
edited May 6, 2015 by Sammi

Contd...

Step 3:

Evaluate function at the end points.

The function is image on the interval image.

Substitute image in .

image

image.

Substitute image in .

image

image.

The maximum value of the function is at image.

The absolute maximum is image.

The minimum value of the function is at image.

The absolute minimum is image.

Solution:

The absolute maximum is image.

The absolute minimum is image.

0 votes

(4)

Step 1:

The function is image, on the interval image.


Evaluate the critical points.

The function is image.

Differentiate image on each side with respect to .

image

image.

Find the critical points, by equate image to zero.

image

image

Solution of the equation in the interval is image.

The critical point is image.

Step 2:

Absolute extrema of a function exist either at the end points or at the critical points.

Substitute the critical point in the function.

Substitute image in image.

image

image.

answered May 6, 2015 by Sammi Mentor
edited May 6, 2015 by Sammi

Contd...

Step 3:

Evaluate function at the end points.

The function is image on the interval image.

Substitute image in image.

image

image.

Substitute image in image.

image

image.

The maximum value of the function is at image.

The absolute maximum is image.

The minimum value of the function is at  image.

The absolute minimum is image.

Solution:

The absolute maximum is image.

The absolute minimum is image.

0 votes

(5)

Step 1:

The function is image and the point is image.

Apply derivative on each side with respect to image.

image

image.

Find the slope of a tangent at the point image.

Substitute image in image.

image

Slope of a tangent line is image.

Step 2:

Find the tangent line equation.

Point - slope form of line equation is image.

Substitute the values image and image in point slope form.

image

image

image

image.

The tangent line equation is image.

Solution:

The tangent line equation is image.

answered May 6, 2015 by Sammi Mentor

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