class 12

Math

Calculus

Application of Derivatives

Suppose the cube $x_{3}px+q$has three distinct real roots where $p>0$and $q>0$. Then which one of the following holds?

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Find the value of $n∈N$ such that the curve $(ax )_{n}+(by )_{n}=2$ touches the straight line $ax +by =2$ at the point $(a,b)˙$

If $f(x)andg(x)$ are continuous functions in $[a,b]$ and are differentiable in$(a,b)$ then prove that there exists at least one $c∈(a,b)$ for which. $∣f(a)f(b)g(a)g(b)∣=(b−a)∣∣ f(a)f_{prime}(c)g(a)g_{prime}(c)∣∣ ,wherea<c<b˙$

Show that between any two roots of $e_{−x}−cosx=0,$ there exists at least one root of $sinx−e_{−x}=0$

The acute angle between the curves $y=∣∣ x_{2}−1∣∣ $and $y=∣∣ x_{2}−3∣∣ $ at their points of intersection when when x> 0, is

If $t$ is a real number satisfying the equation $2t_{3}−9t_{2}+30−a=0,$ then find the values of the parameter $a$ for which the equation $x+x1 =t$ gives six real and distinct values of $x$ .

If the equation $ax_{2}+bx+c=0$ has two positive and real roots, then prove that the equation $ax_{2}+(b+6a)x+(c+3b)=0$ has at least one positive real root.

Find the minimum value of $∣x∣+∣∣ x+21 ∣∣ +∣x−3∣+∣∣ x−25 ∣∣ ˙$

Discuss the extremum of $f(x)={1+sinx,x<0x_{2}−x+1,x≥0atx=0$