Question #1917

From the graph of f(x)=3x^4-20x^3+24x^2, identify on the interval [-1,5] the x-y coordinates of the absolute maximum.
Give your answer in the form (a,b). If there is more than one answer enter points separated by commas. For example (a,b), (c,d)

Expert's answer

Let's find the first derivative of f(x):

f'(x) = 12 x^{3} - 60 x^{2} + 48 x

12 x^{3} - 60 x^{2} + 48 x = 0

x_{1} = 0

x^{2} - 5x + 4 = 0

x_{2} = 4

x_{3} = 1

f'(x) > 0 at (0, 1) and (4,5]

f'(x) < 0 at [-1, 0) and (1,4)

x = 1 - is relative maximum

f(1) = 7

At the bounds

at f(-1) = 47

f(5) = -25

Thus the absolute maximum is (-1, 47)

f'(x) = 12 x

12 x

x

x

x

x

f'(x) > 0 at (0, 1) and (4,5]

f'(x) < 0 at [-1, 0) and (1,4)

x = 1 - is relative maximum

f(1) = 7

At the bounds

at f(-1) = 47

f(5) = -25

Thus the absolute maximum is (-1, 47)

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