Find the intervals in which the function f given by f(x) is(i) Increasing(ii) Decreasing

y = x3 + x-3

Now to find the intervals in which f(x) increases and decreases we first have to find the critical points

Critical points are those points or values of x at which f’(x) = 0

Hence differentiate y with respect to x and equate it to 0

x6 – 1 = 0

x6 = 1

By observation x satisfies both 1 and -1 hence

x = -1 and x = 1

Now we have to check for intervals (-∞, -1), (-1, 1) and (1, ∞)

Case1:

When x < -1 that is x in (-∞, -1)

Observe that x6 is positive and greater than 1 so (x6 – 1) is also positive and the denominator x4 is also positive because of even power

Hence hence f(x) is increasing

Case 2:

When -1 < x < 1 that is x in (-1, 1)

Observe that x6 is positive but less than 1 so (x6 – 1) is negative and the denominator x4 is positive because of even power hence is negative because of (x6 – 1)

Hence hence f(x) is decreasing

Case 3:

When x > 1 that is x in (1, ∞)

Observe that x6 is positive and greater that 1 so (x6 – 1) is positive and the denominator x4 is positive because of even power

Hence hence f(x) is increasing

Hence the conclusion of case1, case2 and case3 is that f(x) is increasing in the interval (-∞, -1) U (1, ∞) and decreasing in the interval (-1, 1)

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