If f(x)=x^3+a x^2+b x+5 sin ^2 x is an increa | vedclass

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(B) Given $f(x) = x^3 + ax^2 + bx + 5 \sin^2 x$ is an increasing function on $R$,so $f'(x) \ge 0$ for all $x \in R$. $f'(x) = 3x^2 + 2ax + b + 10 \sin

Vedclass · Accepted Answer

$a^2-3 b+15 < 0$

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(B) Given $f(x) = x^3 + ax^2 + bx + 5 \sin^2 x$ is an increasing function on $R$,so $f'(x) \ge 0$ for all $x \in R$. $f'(x) = 3x^2 + 2ax + b + 10 \sin x \cos x = 3x^2 + 2ax + b + 5 \sin 2x$. For $f'(x) \ge 0$,the minimum value of $f'(x)$ must be $\ge 0$. Since $\sin 2x$ ranges from $-1$ to $1$,the minimum value of $5 \sin 2x$ is $-5$. Thus,we require $3x^2 + 2ax + b - 5 \ge 0$ for all $x \in R$. For a quadratic $Ax^2 + Bx + C \ge 0$ to hold for all $x$,we must have $A > 0$ and the discriminant $D = B^2 - 4AC \le 0$. Here $A = 3$,$B = 2a$,and $C = b-5$. $D = (2a)^2 - 4(3)(b-5) \le 0$. $4a^2 - 12(b-5) \le 0$. Dividing by $4$,we get $a^2 - 3(b-5) \le 0$. $a^2 - 3b + 15 \le 0$. Since the options provided are strict inequalities,the correct condition is $a^2 - 3b + 15 < 0$.

If $f(x)=x^3+a x^2+b x+5 \sin ^2 x$ is an increasing function on $R$,then

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