Home
Class 12
MATHS
Prove that for any two numbers x(1) and ...

Prove that for any two numbers `x_(1) and x_(2)`
`(2e^(x)+e^(x))/(3)gte(2x_(1)+x_(2))/(3)`

Text Solution

Verified by Experts

consider functionn f(x) =`e^(x)`
Graph of the function is concave upward.
Now consider two points `ax_(1),e^(x_(1))` and `B(x_(2)),e^(x_(2))` on the graph of function.
Let c Be another point which divides AB internaly in ratio 1:2
`therefore` corrdinate of point C are `(2x_(1)+x_(2))/(3),2e^(x_(1))+e^(x_(2))/(3)`
Since graph is concave upward,
`f(2x_(1)+x_(2))/(3)lt2e^(x_(1))+(e^(x_(2))/(3))`
or `(2x_(1)+x_(2))/(3)lt (e^(2x_(1))+e^(x_(2))/(3))`
Promotional Banner

Topper's Solved these Questions

  • MONOTONICITY AND MAXIMA MINIMA OF FUNCTIONS

    CENGAGE PUBLICATION|Exercise Solved Examples|20 Videos

  • MONOTONICITY AND MAXIMA MINIMA OF FUNCTIONS

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 6.1|10 Videos

  • METHODS OF DIFFERENTIATION

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer Type|7 Videos

  • MONOTONOCITY AND NAXINA-MINIMA OF FUNCTIONS

    CENGAGE PUBLICATION|Exercise Comprehension Type|6 Videos

Similar Questions

Explore conceptually related problems

Prove that for any two numbers x_1 and x_2.(2e^(x_1)+e^(x_2))/3gte^((2(x_1)+x_2)/3)

int e^(x).(x^(2)+1)/((x+1)^(2))dx

int (e^(4x)-2e^(3x)+5e^(x)-2)/(e^(x)+1)dx

Evaluate int(e^(2x)-2e^(x))/(e^(2x)+1)dx

int_(0)^(1)e^(2x)e^(e^(x) dx

Evaluate : int (2e^(4x)-3e^(2x)+4)/(e^(3x))dx

int e^(x)((1-x)/(1+x^(2)))^(2)dx

Find (dy)/(dx) : y =2^(x+2)-e^(x+1)+3log_(e) x

Evaluate : int ((e^(5x)-2e^(3x)+3))/(e^(x)) dx

Prove that, int (e^(x))/(x^(4))dx=-(e^(x))/(3)[(1)/(x^(3))+(1)/(2x^(2))+(1)/(2x)]+(1)/(6)int (e^(x))/(x)dx .