Class 12

Math

Calculus

Continuity and Differentiability

Prove that the function $f(x)=5x−3$is continuous at $x=0$, at $x=−3$and at $x=5$.

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Let $f$ be a continuous function defined onto [0,1] with range [0,1]. Show that there is some $c$ in [0,1] such that $f(c)=1−c$

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

Discuss the continuity and differentiability of $f(x)=∣x+1∣+∣x∣+∣x−1∣∀x∈R;$ also draw the graph of $f(x)$

The right hand derivative of $f(x)=[x]tanπxatx=7$ is (where [.] denotes the greatest integer function) $0$ b. $7π$ c. $−7π$ d. none of these

Which of the function is non-differential at $x=0?$ $f(x)=∣∣ x_{3}∣∣ $

let $f(x)=x_{3}−x_{2}−3x−1,g(x)=(x+1)a$and $h(x)=g(x)f(x) $ where $h$ is a rational function such that $(1)$ it is continuous everywhere except when $x=−1,(2)x→∞lim h(x)=∞$ and $(3)x→−1lim h(x)=21 $ then the value of $h(1)$

Discuss the continuity of $f(x)=∣x∣sgn(x_{3}−x)$

Discuss the differentiability of $f(x)=max{tan_{−1}x,cot_{−1}x}˙$