Find, if it exists, a value c in the interval [1, 4] such that the instantaneous rate of change of f(x) = 12 at c is the same as the average rate of change of f over the interval [1, 4]. (If an answer does not exist, enter DNE.)

  c = 2.25

Step-by-step explanation:

f(x) is a continuous and differentiable function on the interval, so the Mean Value Theorem guarantees a value for c exists.

The average slope is ...

  m = (f(4) -f(1))/(4 -1) = (24 -12)/3 = 4

The point at which the derivative is 4 is ...

  f'(c) = 6/√c = 4

  √c = 6/4

  c = 2.25

maximum area = xy = (16)(32) = 512 ft2

to maximize the area of the garden, the side of the fence parallel to the side of the house should be 32 ft long, and the other two sides should both have length 16 ft.'

source

x= -3 and

step-by-step explanation:

to find the roots of a cubic, use division or factoring to find the factors then set equal to 0. as this is a difference of cubes, it factors into

where a is the cube root of .

here   this is 27 and the cube root of 27 is 3. so we write using the form:

.

set each factor to 0 and solve for x.

x-3=0

x=3

the quadratic expression does not factor and must be solved using the quadratic formula.

where a = 1, b=3, and c=9

this cubic has two complex roots.