Answer in detail please Find the critical points of the function below. Find and

Question

Answer in detail please

Find the critical points of the function below. Find and classify (categorize the point as maximum, minimum or inflection point) of the function below. Find and classify (categorize the point as maximum, minimum or inflection point) of the function below. Determine the concavity and relative extrema of the function below. Assume that f(x) below shows the profit in dollars when x units of goods are produced. What are the 1) maximum profit and 2) maximum units of production possible?

Explanation / Answer

Ans1. just differentiate the fucntion and equate it to zero

it gives..3x^2 + 6x = 0..i.e x = 0; -2..these are critical points

Ans2. find forst critical points as we did above

gives..-3x^2-6x=0..x=0;-2

now again differentiate it..gives..f''(x) = -6x-6=0

if f''(a)=0, then point 'a' is inflection point; if f''(a)<0, then it is maxima; if f''(a)>0, then it is minima.

at x=0, f''(0)=-6<0, therefore it is maxima point

at x=-2, f''(-2) = 6>0, it is minima.

Ans3. f(x)=x^5; f'(x) = 5x^4; f''(x) = 20x^3

critical point is x=0, and f''(0)=0, so it point of inflection.

Ans4. A function is concave at its point of minima

so we find it here too; f'(x) = 4x^3 - 32 = 0, gives x^3 - 8=0..so x=2;

f''(x) = 12x^2; so f''(2)=48>0, so it is minima point; hence there is concavity in this curve(as it is minima)

Ans5. f'(x) = -4.5x^2 + 19x +10 = 0; it gives x= -0.47; 4.69

f''(x) = -9x+19; minima at x=-0.47 and maxima at 4.69

so calculate maximum profit and minimum profit at these two points.

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