If f(x) = x5 - 20x3 + 240 x, then f(x) isa)monotonically decreasing everywhereb)monotonically decreasing on (0, ∞)c)monotonically increasing everywhered)monotonically increasing only in (-∞, 0)Correct answer is option 'C'. Can you explain this answer?

Question

If f(x) = x5 - 20x3 + 240 x, then f(x) isa)monotonically decreasing everywhereb)monotonically decreasing on (0, &infin;)c)monotonically increasing everywhered)monotonically increasing only in (-&infin;, 0)Correct answer is option 'C'. Can you explain this answer?

Answer

To determine whether f(x) is monotonically decreasing or increasing, we need to find the derivative of f(x) and analyze its sign.

Given: f(x) = x^5 - 20x^3 + 240x

Taking the derivative of f(x) with respect to x:
f'(x) = 5x^4 - 60x^2 + 240

To find the critical points, we set f'(x) equal to zero and solve for x:
5x^4 - 60x^2 + 240 = 0

We can factor out a common factor of 5:
5(x^4 - 12x^2 + 48) = 0

Using the quadratic formula to solve for x^2, we get:
x^2 = (12 ± √(12^2 - 4(1)(48))) / 2
x^2 = (12 ± √(144 - 192)) / 2
x^2 = (12 ± √(-48)) / 2

Since we have a negative value under the square root, there are no real solutions for x^2, which means there are no critical points.

Since there are no critical points, we can conclude that the derivative f'(x) is either always positive or always negative.

To determine the sign of f'(x), we can use the second derivative test. Taking the derivative of f'(x):
f''(x) = 20x^3 - 120x

To find the critical points of f''(x), we set f''(x) equal to zero and solve for x:
20x^3 - 120x = 0
20x(x^2 - 6) = 0

Setting each factor equal to zero:
20x = 0, which gives x = 0
x^2 - 6 = 0, which gives x = ±√6

We now have three critical points: x = 0, x = √6, and x = -√6. However, since these critical points are not within the domain of the function f(x), we discard them.

Since there are no critical points within the domain of f(x), we can conclude that f'(x) is always positive or always negative.

Therefore, f(x) is either monotonically increasing everywhere or monotonically decreasing everywhere.

The answer is:
a) monotonically decreasing everywhere
b) monotonically increasing everywhere

Answer

Solution : c)

Answer

Solution : c)

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