Answer to Question #1917 in Calculus for Bujju

Question

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)

Answer 1

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

f'(x) = 12 x³ - 60 x² + 48 x
12 x³ - 60 x² + 48 x = 0
x₁ = 0
x² - 5x + 4 = 0
x₂ = 4
x₃ = 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)