ncert-solutions-class-12-chapter-5-continuity-and-differentiability-miscellaneous-exercise-question-20

20. Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.


Continuity and Differentiability

Miscellaneous Exercise

ncert solutions class 12 chapter 5 continuity and differentiability miscellaneous exercise question 20

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Class 12

Continuity and Differentiability

Miscellaneous Exercise

Differentiate w.r.t. x the function in Exercises 1 to 11.

1. (left(3x^2-9x+5right)^9).

2. ({sin}^3{x}+{cos}^6{x}).

3. (left(5xright)^{3cos{2}x}).

4. ({sin}^{-1}{left(xsqrt xright)}),(0le xle1).

5. (frac{{cos}^{-1}{frac{x}{2}}}{sqrt{2x+7}}),(-2<x<2).

6. ({cot}^{-1}{left[frac{sqrt{1+sin{x}}+sqrt{1-sin{x}}}{sqrt{1+sin{x}}-sqrt{1-sin{x}}}right]}),(0<x<frac{pi}{2}).

7. (left(log{x}right)^{log{x}}), (x>1).

8. (cos{left(acos{x}+bsin{x}right)}), for some constant a and b.

9. (left(sin{x}-cos{x}right)^{left(sin{x}-cos{x}right)}), (frac{pi}{4}<x<frac{3pi}{4}).

10. (x^x+x^a+a^x+a^a), for some fixed a>0 and x>0.

11. (x^{x^2-3}+left(x-3right)^{x^2}), for x>3.

12. Find (frac{dy}{dx}), if (y=12left(1-cos{t}right)),(x=10left(t-sin{t}right)),(-frac{pi}{2}<t<frac{pi}{2}).

13. Find (frac{dy}{dx}), If (y={sin}^{-1}{x}+{sin}^{-1}{sqrt{1-x^2}}),(-1le xle1).

14. If (xsqrt{1+y}+ysqrt{1+x}=0), for (-1<x<1), prove that (frac{dy}{dx}=-frac{1}{left(1+xright)^2}).

15. If (left(x-aright)^2+left(y-bright)^2=c^2), for some c>0, prove that (frac{left[1+left(frac{dy}{dx}right)^2right]^frac{3}{2}}{frac{d^2y}{dx^2}}) is a constant independent of a and b.

16. If (cos{y}=xcos{left(a+yright)} ) with (cos{a}neqpm1), prove that (frac{dy}{dx}=frac{{cos}^2{left(a+yright)}}{sin{a}}).

17. If (x=aleft(cos{t}+tsin{t}right) ) and (y=aleft(sin{t}-tcos{t}right)), find (frac{d^2y}{dx^2}).

18. If (fleft(xright)=left|xright|^3), show that f”(x) exists for all real x and find it.

19. Using the fact that (sin{(A+B)}=sin{A}cos{B}+cos{A}sin{B}) and the differentiation, obtain the sum formula for cosines.

20. Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.

21. If (y=left|begin{matrix}fleft(xright)&gleft(xright)&hleft(xright)\l&m&n\a&b&c\end{matrix}right|), prove that (frac{dy}{dx}=left|begin{matrix}f'(x)&g'(x)&h'(x)\l&m&n\a&b&c\end{matrix}right|).

22. If (y=e^{a{cos}^{-1}{x}}), (-1le xle1), show that (left(1-x^2right)frac{d^2y}{dx^2}-xfrac{dy}{dx}-a^2y=0).



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