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Using calculus, show that the maximum va...

Using calculus, show that the maximum value of the function `(1/x)^x` is `e^(1/e)`.

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UNITED BOOK HOUSE-QUESTION PAPER 2017-EXERCISE
  1. Solve: tan^(-1)(x+1)+tan^(-1)(x-1)=tan^(-1)(8/31).

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  2. If A=([2,-1],[1,3]) then show that A^2-5A+7I2=0 hence find A^(-1)

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  3. Solve by Cramer's rule: x+3y= 4,y+3z = 7, 4x+z= 6.

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  4. Prove that |[2ab,a^2,b^2],[a^2,b^2,2ab],[b^2,2ab,a^2]|=-(a^3+b^3)^2.

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  5. If f(x)= ((a+x)/(b+x))^(a+b+2x) then prove that f'(0)= [2log(a/b)+(b...

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  6. Evaluate: int (xdx)/(x^4-x^2+1)

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  7. Solve: ydx-(x+2y^2)dy=0.

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  8. In a certain culture the rate of increment of bacterial at any instant...

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  9. The position vectors of four points A,B,C and D are 4hati+8 hat j+12ha...

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  10. If a vector 2 i ^ +3 j ^ ​ +8 k ^ is perpendicular to the vecto...

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  11. Evaluate: overset(pi/4)underset(0)intlog(1+tan theta)d theta.

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  12. find the value of overset(2)underset(1)int 5x^2dx.

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  13. A man is known to speak the truth 3 out of 4 times. HE throws an unbia...

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  14. If the sum of the mean and variance of a binomial distribution for 5 r...

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  15. Solve the following linear programming problem by graphical method and...

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  16. Using calculus, show that the maximum value of the function (1/x)^x is...

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  17. Using integration, prove that the area of the closed region bounded by...

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  18. Solve: xdy-ydx= sqrt(x^2+y^2)dx.

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  19. Find the shortest distance between the straight lines vecr = -4hati+...

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  20. A variable plane is at a constant distance p from the origin and meets...

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