Class 12

Math

Calculus

Application of Derivatives

Find the approximate value of $f(5.001)$, where $f(x)=x_{3}−7x_{2}+15$.

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Find the angle between the curves $x_{2}−3y_{2} =a_{2}andC_{2}:xy_{3}=c$

Let $f(x)=xa +x_{2}˙$ If it has a maximum at $x=−3,$ then find the value of $a˙$

Prove that if $2a02<15a,$ all roots of $x_{5}−a_{0}x_{4}+3ax_{3}+bx_{2}+cx+d=0$ cannot be real. It is given that $a_{0},a,b,c,d∈R˙$

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $R$ is $3 2R $

If $2a+3b+6c=0,$ then prove that at least one root of the equation $ax_{2}+bx+c=0$ lies in the interval (0,1).

Let $x$ be the length of one of the equal sides of an isosceles triangle, and let $θ$ be the angle between them. If $x$ is increasing at the rate (1/12) m/h, and $θ$ is increasing at the rate of $180π $ radius/h, then find the rate in $m_{3}$ / $h$ at which the area of the triangle is increasing when $x=12mandthη=π/4.$

On the curve $x_{3}=12y,$ find the interval of values of $x$ for which the abscissa changes at a faster rate than the ordinate?

If $fogoh(x)$ is an increasing function, then which of the following is not possible? (a)$f(x),g(x),andh(x)$ are increasing (b)$f(x)andg(x)$ are decreasing and $h(x)$ is increasing (c)$f(x),g(x),andh(x)$ are decreasing