Increasing function

Calculus Level 3

How many integers a a are there such that f ( x ) = ( x 2 + a x + 2 a ) e x f(x)=(x^2+ax+2a)e^x is a strictly increasing function in ( , ) ? (-\infty , \infty)?


The answer is 7.

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2 solutions

Sauresh Bhowmick
May 8, 2014

f ( x ) = e x ( x 2 + a x + 2 a ) f(x)=e^x(x^2+ax+2a) , so, f ( x ) = e x ( x 2 + ( a + 2 ) x + 3 a f'(x)= e^x(x^2+(a+2)x+3a . For f ( x ) f(x) to be increasing f ( x ) 0 f'(x)\ge 0 . That is e x ( x 2 + ( a + 2 ) x + 3 a ) 0 e^x(x^2+(a+2)x+3a) \ge 0 x 2 + ( a + 2 ) x + 3 a ) 0 \implies x^2 +(a+2)x+3a) \ge 0 as e x 0 e^x \ge 0 , and ( x + ( a + 2 ) / 2 ) 2 + ( 3 a ( ( a + 2 ) / 2 ) 2 ) 0 (x+(a+2)/2)^ 2 +(3a-((a+2)/2)^ 2)\ge 0 , which leads us to 12 a ( a + 2 ) 2 12a \ge ( a+2)^ 2 a 2 8 a + 4 0 \implies a^2-8a+4 \le 0 0.53 a 7.46 \implies 0.53 \le a \le 7.46 , Since a a is integer, so number of possible values of a a is 7 \boxed{7} .

@sauresh bhowmick , I have done up the LaTex for you. Hope that you can learn up fast.

Chew-Seong Cheong - 4 years, 10 months ago

Could u pls explain me how u did the squaring part and then dividing it by 2

rit tak - 3 years, 4 months ago

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That was completing the square technique. There's a wiki page on it.

A better explanation would be why this needed to be done...

Anton Dubovik - 3 years, 2 months ago

f ( x ) = ( x 2 + a x + 2 a ) e x f ( x ) = ( x 2 + a x + 2 a ) e x + ( 2 x + a ) e x = ( x 2 + ( a + 2 ) x + 3 a ) e x \begin{aligned} f(x) & = (x^2+ax+2a)e^x \\ f'(x) & = (x^2+ax+2a)e^x + (2x+a)e^x \\ & = \left(x^2+(a+2)x+3a\right)e^x \end{aligned}

For f ( x ) f(x) to be increasing, f ( x ) 0 f'(x) \ge 0 . Since e x > 0 e^x > 0 for all x x , we need g ( x ) = x 2 + ( a + 2 ) x + 3 a 0 g(x) = x^2+(a+2)x + 3a \ge 0 . That is when g ( x ) g(x) has no real roots or

( a + 2 ) 2 12 a 0 a 2 + 4 a + 4 12 a 0 a 2 8 a + 4 0 ( a 4 + 2 3 ) ( a 4 2 3 ) 0 \begin{aligned} (a+2)^2 - 12a & \le 0 \\ a^2+4a+4 - 12a & \le 0 \\ a^2 - 8a + 4 & \le 0 \\ (a-4+2\sqrt 3)(a-4-2\sqrt 3) & \le 0 \end{aligned}

4 2 3 a 4 + 2 3 0.534 a 7.464 \implies 4-2\sqrt 3 \le a \le 4+2\sqrt 3 \approx 0.534 \le a \le 7.464 or for integers 1 a 7 1 \le a \le 7 , a total of 7 \boxed{7} integral a a .

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