Consider the function below. (Round the answer to two decimal places.)
f(x)=(x+7)/(sqrt(x^2 +7))
a) Find the maximum value.
I believe the answer to this is f(x)=2.83. Is that right?
b) Find the value of x at which f increases most rapidly.
I think it has something to do with finding the derivative, but I'm not sure where to go from there.
f(x)=(x+7)/(sqrt(x^2 +7))
a) Find the maximum value.
I believe the answer to this is f(x)=2.83. Is that right?
b) Find the value of x at which f increases most rapidly.
I think it has something to do with finding the derivative, but I'm not sure where to go from there.
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f (x) = (x+7) / √(x²+7)
f '(x) = (7 - 7x) / (x²+7)^(3/2) (fully simplified)
Through the first derivative test you would find that x=1 is a critical number, with f ' > 0 on (-∞,1) and f ' < 0 on (1,∞), implying there's a max at x=1.
f(1) = 8/√8 ≈ 2.83, so you're right with (a).
(b) seems to be asking you what the highest rate of change of f is, so you find the max of the derivative. To do this, you find the second derivative
f "(x) = -(21(2x³ - 4x² - 21x + 7))/(x² + 7)^(7/2)
Apply the derivative test to the second derivative to find the max of f.
f '(x) = (7 - 7x) / (x²+7)^(3/2) (fully simplified)
Through the first derivative test you would find that x=1 is a critical number, with f ' > 0 on (-∞,1) and f ' < 0 on (1,∞), implying there's a max at x=1.
f(1) = 8/√8 ≈ 2.83, so you're right with (a).
(b) seems to be asking you what the highest rate of change of f is, so you find the max of the derivative. To do this, you find the second derivative
f "(x) = -(21(2x³ - 4x² - 21x + 7))/(x² + 7)^(7/2)
Apply the derivative test to the second derivative to find the max of f.