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Elena L [17]
3 years ago
10

Find the critical points, domain endpoints, and local extreme values for the function

Mathematics
1 answer:
aliina [53]3 years ago
8 0

Answer:

a) x = -3, b) x = 0, x = -6, c) x = 0, d) x = -6

Step-by-step explanation:

a) Let derive the function:

f'(x) = \frac{10\cdot x \cdot (x+3)-5\cdot x^{2}}{25\cdot (x+3)^{2}}

f'(x) is undefined when denominator equates to zero. The critical point is:

x = -3

b) f'(x) = 0 when numerator equates to zero. That is:

10\cdot x \cdot (x+3) - 5\cdot x^{2} = 0

10\cdot x^{2}+30\cdot x -5\cdot x^{2} = 0

5\cdot x^{2} + 30\cdot x = 0

5\cdot x \cdot (x+6) = 0

This equation shows two critical points:

x = 0, x = -6

c) The critical points found in point b) and the existence of a discontinuity in point a) lead to the conclusion of the existence local minima and maxima. By plotting the function, it is evident that x = 0 corresponds to a local maximum. (See Attachment)

d) By plotting the function, it is evident that x = -6 corresponds to a local minimum. (See Attachment)

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