Abs. Max and Min

Question

Find the absolute maximum and minimum of f(x) = x^(1/2) ((x^(2)/5)-4) on the interval [0,4]. Give clear steps and explanations

Answer 1

f(x) = x^5/2/5 - 4x^1/2

To find the relative max and min, take the derivative, set to zero and solve for x

f'(x) = 5x^3/2/10 - 2/√x = (x²-4)/(2√x)

x²-4 = 0

x = ±2 (-2 not in [0,4])

To find the absolute max and min, find the values of f(0), f(4), f(2) and pick max and min

f(0) = 0

f(4) = 2(16/5-4) = 32/5-8 = -8/5

f(2) = √2(4/5-4) = √2(-4) = -4.525

Absolute max = 0

Absolute min = -4√2