Question

Find any relative extrema of the function. List each extremum along with the​ x-value at which it occurs. Identify intervals over which the function is increasing and over which it is decreasing. Then sketch a graph of the function.

f​(x)=7−3x+x2 Question content area bottom
Part 1
Describe any relative extrema. Select the correct choice below​ and, if​ necessary, fill in the answer​ box(es) to within your choice.
A.The relative minimum point(s) is/are _____ and there are no relative maximum points.
​(Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as​ needed.)
B.The relative minimum​ point(s) is/are ___ and the relative maximum​ point(s) is/are _____ ​(Simplify your answers. Type ordered​ pairs, using integers or fractions. Use a comma to separate answers as​ needed.)
C.The relative maximum point(s) is/are ______ and there are no relative minimum points.
​(Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as​ needed.)
D. There are no relative minimum points and there are no relative maximum points.

Answer 1

A. The relative minimum point is (3/2, f(3/2)).

To find the relative extrema of the function f(x)=7−3x+x^2, we need to find the critical points by setting the derivative equal to zero.

Taking the derivative of f(x), we get:

f’(x) = -3 + 2x

Setting f’(x) equal to zero, we have:

-3 + 2x = 0

Solving for x, we find:

2x = 3

x = 3/2

The critical point occurs at x = 3/2.

To determine the type of extrema at this critical point, we can use the second derivative test. Taking the derivative of f’(x), we get:

f’’(x) = 2

Since the second derivative is positive, f’’(x) > 0, the function has a relative minimum at x = 3/2.

Therefore, the correct choice is:

A. The relative minimum point is (3/2, f(3/2)).