Find the intervals in which the following functions are strictly increasing or decreasing:
(a) $x^{2} + 2 x - 5$
(b) $10 - 6 x - 2 x^{2}$
(c) $- 2 x^{3} - 9 x^{2} - 12 x + 1$
(d) $6 - 9 x - x^{2}$
(e) $f (x) = (x + 1)^{3} (x - 3)^{3}$

Question

Find the intervals in which the following functions are strictly increasing or decreasing: (a) x2+2x−5(b)  10−6x−2x2(c) −2x3−9x2−12x+1(d) 6−9x−x2(e) f(x)=(x+1)3(x−3)3

Filo · Accepted Answer

(a)We have,f(x)=x2+2x−5∴f′(x)=2x+2Now, f′(x)=0⇒x=−1The point x=−1 divides the real line into two disjoint intervals i.e.,  (−∞,−1) and (−1,∞).In interval (−∞,−1),f′(x)=2x+20Thus, f is strictly increasing for x>1.  (b)f(x)=10−6x−2x2∴ f′(x)=−6−4xNow, f′(x)=0⇒x=−23​The point x =−23​ divides the real line into two disjoint intervals i.e.,  (−∞,−23​) and (−23​,∞ ).In interval (−∞,−23​)  i.e., when  x0 Thus f is strictly increasing in this interval. (c)f(x)=−2x3−9x2−12x+1∴f′(x)=−6x3−18x2−12=−6(x2+3x+2)=−6(x+1)(x+2)Now,f′(x)=0⇒x=−1 and x=−2Points x=−1 and x=−2 divide the real line into three disjoint intervalsi.e., (−∞,−2 )(−2,−1 ) and (−1,−∞).In intervals  (−∞,−2 ) and (1,−∞)  i.e., when x−1, f′(x)=−6(x+1)(x+2)−1. Now, in interval (−2,−1) i.e., when −20∴ f is strictly increasing for  −20⇒2x+90⇒x>−29​(e) We have,f(x)=(x+1)3(x−3)3 f′(x)=3(x+1)2(x−3)3+3(x−3)2(x+1)3=3(x+1)2(x−3)2[x−3+x+1]=3(x+1)2(x−3)2(2x−2)=6(x+1)2(x−3)2(x−1)Now,f′(x)=0⇒x=−1,3,1The points x=−1,x=1, and x=3 divide the real line into four disjoint intervalsi.e.,(−∞,−1),(−1,1),(1,3) and (3,∞).In intervals (−∞,−1) and (−1,1), f′(x)=6(x+1)2(x−3)2(x−1)0∴ f is strictly increasing in intervals (1,3) and (3,∞).

Question Text	Find the intervals in which the following functions are strictly increasing or decreasing: (a) $x^{2} + 2 x - 5$ (b) $10 - 6 x - 2 x^{2}$ (c) $- 2 x^{3} - 9 x^{2} - 12 x + 1$ (d) $6 - 9 x - x^{2}$ (e) $f (x) = (x + 1)^{3} (x - 3)^{3}$
Answer Type	Text solution:1
Upvotes	150

Find the intervals in which the following functions are strictly increasing or decreasing:
(a) $x^{2} + 2 x - 5$
(b) $10 - 6 x - 2 x^{2}$
(c) $- 2 x^{3} - 9 x^{2} - 12 x + 1$
(d) $6 - 9 x - x^{2}$
(e) $f (x) = (x + 1)^{3} (x - 3)^{3}$

Text solutionVerified

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