IPMAT Question Paper 2019 | IPM Indore Quants

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Question 39: The function f(x) = \\frac{x^{3} - 5x^{2} - 8x}{3}\\) is

1. positive and monotonically increasing for x \\in (-\infty, \frac{5-\sqrt{57}}{2}\\)) and x \\in (\frac{5+\sqrt{57}}{2}, +\infty\\))
2. negative and monotonically decreasing for x \\in (-\infty, \frac{5-\sqrt{57}}{2}\\) and x \\in (\frac{5+\sqrt{57}}{2},+\infty\\))
3. negative and monotonically increasing for x \\in (-\infty, \frac{5-\sqrt{57}}{2}\\)) and positive and monotonically increasing for x \\in (\frac{5+\sqrt{57}}{2},+\infty\\))
4. positive and monotonically increasing for x \\in (-\infty, \frac{5-\sqrt{57}}{2}\\)) and negative and monotonically decreasing for x \\in (\frac{5+\sqrt{57}}{2},+\infty\\))

Video Explanation

Explanatory Answer

Let’s find the roots of the function, That is where the function takes the value 0.

f(x) = 0.
\\frac{x^{3} - 5x^{2} - 8x}{3}\\) = 0

x³- 5x²- 8x = 0
x (x²−5x−8) = 0

x = 0 or x²−5x−8 = 0

x²−5x−8 = 0

x²−5x + \\frac{25}{4}\\) −\\frac{25}{4}\\) −8 = 0
(x - \\frac{5}{2}\\) )²- \\frac{57}{4}\\) = 0

(x - \\frac{5}{2}\\) )²= \\frac{57}{4}\\)
x - \\frac{5}{2}\\) = +\\frac{\sqrt{57}}{2}\\) (or) x - \\frac{5}{2}\\) = - \\frac{\sqrt{57}}{2}\\)

x = \\frac{5}{2}\\) + \\frac{\sqrt{57}}{2}\\) or x = \\frac{5}{2}\\) - \\frac{\sqrt{57}}{2}\\)

Therefore, the roots of the equation \\frac{x^{3} - 5x^{2} - 8x}{3}\\) are 0, \\frac{5}{2}\\) + \\frac{\sqrt{57}}{2}\\), \\frac{5}{2}\\) - \\frac{\sqrt{57}}{2}\\)

That means, when plotted f(x) will be 0 at 3 points when x = 0 or \\frac{5}{2}\\) + \\frac{\sqrt{57}}{2}\\) or \\frac{5}{2}\\) - \\frac{\sqrt{57}}{2}\\)



Arranged in increasing order the roots are \\frac{5}{2}\\) - \\frac{\sqrt{57}}{2}\\) , 0 , \\frac{5}{2}\\) + \\frac{\sqrt{57}}{2}\\).

Check f(x) at a point which is less than \\frac{5}{2}\\) - \\frac{\sqrt{57}}{2}\\) and at a point greater than \\frac{5}{2}\\) + \\frac{\sqrt{57}}{2}\\)

f(-2) = \\frac{(-2)^{3} - 5(-2)^{2} - 8(-2)}{3}\\) = \\frac{-8-20-16}{3}\\) = \\frac{-44}{3}\\) = -ve.
f(10) = \\frac{(10)^{3} - 5(10)^{2} - 8(10)}{3}\\) = \\frac{1000-500-80}{3}\\) = \\frac{420}{3}\\) = 140 =+ve

Since f(x) is negative below \\frac{5}{2}\\) - \\frac{\sqrt{57}}{2}\\) and positive above \\frac{5}{2}\\) + \\frac{\sqrt{57}}{2}\\).

And since it is a continuous function, the function looks something like this.



From the graph, only Option C is correct.

f(x) is negative and monotonically increasing for x \\in (-\infty, \frac{5-\sqrt{57}}{2}\\)) and positive and monotonically increasing for x \\in (\frac{5+\sqrt{57}}{2},+\infty\\))

The question is"The function f(x) = \\frac{x^{3} - 5x^{2} - 8x}{3}\\) is"

Hence, the answer is negative and monotonically increasing for x \\in (-\infty, \frac{5-\sqrt{57}}{2}\\)) and positive and monotonically increasing for x \\in (\frac{5+\sqrt{57}}{2},+\infty\\))

Choice C is the correct answer

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