Question:

If pp and qq are respectively the global maximum and global minimum of the function f(x)=x2e2xf(x) = x^2 e^{2x} on the interval [2,2][-2, 2] , then pe4+qe4=pe^{-4} + qe^4 =

Updated On: Apr 4, 2024
  • 00
  • 4e84e^8
  • 44
  • 4e8+14e^8 + 1
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The Correct Option is C

Solution and Explanation

Given, f(x)=x2e2xf(x)=x^{2} e^{2 x}
Differentiating w.r.t. xx, we get
f(x)=2e2xx2+2xe2x=2e2x(x2+x)f'(x) =2 e^{2 x} x^{2}+2 x e^{2 x}=2 e^{2 x}\left(x^{2}+x\right)
f(x)=02e2x(x2+x)=0\Rightarrow f'(x) =0 \Rightarrow 2 e^{2 x}\left(x^{2}+x\right)=0
e2x=0\Rightarrow e^{2 x} =0 or x2+x=0x=0,1 x^{2}+x=0 \Rightarrow x=0,-1
So, maxima and minima can be attain at either of
2,1,0,2-2,-1,0,2 as the function bound.
f(2)=(2)2e4=4e4\therefore f(-2)=(-2)^{2} e^{-4}=4 e^{-4}
f(1)=(1)2e2f(-1) =(-1)^{2} e^{-2}
f(0)=0f(0) =0
f(2)=(2)2e4=4e4f(2) =(2)^{2} e^{4}=4 e^{4}
p=4e4\therefore p=4 e^{4} and q=0 q= 0
Therefore, pe4+qe4=4e4(e4)+0=4e0p e^{-4}+q e^{4}=4 e^{4}\left(e^{-4}\right)+0=4 e^{0}
=4×1=4=4 \times 1=4
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Concepts Used:

Maxima and Minima

What are Maxima and Minima of a Function?

The extrema of a function are very well known as Maxima and minima. Maxima is the maximum and minima is the minimum value of a function within the given set of ranges.

There are two types of maxima and minima that exist in a function, such as:

  • Local Maxima and Minima
  • Absolute or Global Maxima and Minima