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(a) `a^2-3b+15\ lt0` (b) `a^2-3b+20\ lt0`(c) `a^2-3b+25\ lt0` (d) `a^2-3b+30\ lt0`

Transcript

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00:00 - 00:59 | hello everyone next question satisfy the relation a square - 3 + 15 is less than zero is a - 3 + 2000 the square - 3 x 25 s0nd square - 3 + 30 is less than zero we are given a quadratic function f of x is equal to x cube + x square + bx + 5 sin square x over the real number domain are also given that f is strictly increasing over its domain the real numbers we are suppose to find the relation between a and b now that they quickly increases on its to mean |

01:00 - 01:59 | that is it increases on our that mean that derivative of X with respect to a positive for all of units to mean letters and calculate the derivative 3 X square + 2 X + b + 5 x 2 sin x cos x now we know that 2 sin x cos x can be written as sin 2x to the derivative of f with respect to X is going to be 3 X square + 2 X + b + 5.2 this is going to be strictly greater than zero for all X belonging to are now this expression must always remain positive but we know that the sin can definitely go negative that it was the lowest value of sin 2x so that we can ensure that |

02:00 - 02:59 | the entire expression always greater than zero we know that to main of sin2x is going to negative 121 the closed interval to the minimum value of sin 2x is going to be negative one which me expression become 3 X square + 2 X + b y x negative on or minus 5 is always greater than 0 in the quadratic equation now in order for this expression to quadratic expression to always be positive two conditions have to be met quantico efficient must be positive and determinant of the expression is negative strictly quirino is already positive it is equal to 3 of the determined to be negative that means that |

03:00 - 03:59 | to a whole square minus 4 x b - 5 x 3 is less than zero or four square minus 12 + 60 is less than zero dividing the entire thing by 4 which is not equal to zero a square - 3 + 15 is less than zero let us see which of our options at size and conditioning theory y square - 3 + 15 is less than zero satisfy the given condition therefore we can conclude that option is correct thank you |