Class 12

Math

Calculus

Continuity and Differentiability

Find $dxdy $ in the following:$sin_{2}y+cosxy=π$

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Let $f(x)={xαcotx +x_{2}β 31 0<∣x∣≤1x=0 )$ If $f(x)$ is continuous at $x=0$ then the value of $α_{2}+β_{2}$ is $1$ b. $2$ c. $5$ d. 9

Let $f(x+y)=f(x)+f(y)$ for all $xandy˙$ If the function $f(x)$ is continuous at $x=0,$ show that $f(x)$ is continuous for all $x˙$

Suppose $f$ is a continuous map from $R$ to $R$ and $f(f(a))=a$ for some $a˙$ Show that there is some $b$ such that $f(b)=b˙$

if $x=t_{3}1+t ,y=2t_{2}3 +t2 $ satisfies $f(x)⋅{dxdy }_{3}=1+dxdy $ then $f(x)$ is:

If $f(x)={s∈(cos_{−1}x)+cos(sin_{−1}x),x≤0s∈(cos_{−1}x)−cos(sin_{−1}x,x>0)$ . Then at $x=0$ $f(x)$ is continuous and differentiable $f(x)$ is continuous but not differentiable $f(x)$ not continuous but differentiable $f(x)$ is neither continuous nor differentiable

Let $f$ be a function with continuous second derivative and $f(0)=f_{prime}(0)=0.$ Determine a function $g$ by $g(x)={xf(x) ,x=00,x=0$ Then which of the following statements is correct? $g$ has a continuous first derivative $g$ has a first derivative $g$ is continuous but $g$ fails to have a derivative $g$ has a first derivative but the first derivative is not continuous

If $f(x)={x,x≤1,x_{2}+bx+c,x>1_{′}$ find b and c if function is continuous and differentiable at $x=1$

The function $f:RR$ satisfies $f(x_{2})f˙ _{x}=f_{prime}(x)f˙ _{prime}(x_{2})$ for all real $x˙$ Given that $f(1)=1$ and $f_{1}=8$ , then the value of $f_{prime}(1)+f_{1}$ is $2$ b. $4$ c. $6$ d. 8