Question

Examine the following functions at point x0 0 for differentiability 2 sin,für ± 0, für x 0 x sin, für 0, für 0 - { (a) f(x) (b) f(ax) 0, 0, Note: Consider the limit of the difference quotient. Is the function f (x) = a | x *4 differentiable everywhere with a E R? X Determine all constants a, B E R, so that the following function is differentiable ac ross R f(x) Find the equation of the tangent t to the graph of f (x) In x at x 2

Answer 1

@ f(x) = $ x sint. ** ox=0 Now fo) - sim froth) – from te los haso him n nsinh- O lim sint h-10 hyo similarly rence fee for tio is not limit does not exist limit does not exist differentiable at a=0. o f(x) = { gp sin £ x #0 x=0 R.H.D. fol=hto f'co - lim froth)-fros h tim n? sint -0. tim ht0 hsint 0. ht0 _ n L.H.D. f!2015 im hao tím hyo fola floth) h. 0 - 6 Sivi Lm - Sin1 - 0 , _ ho n L.H.D = R.H.D. at x=0. Mence fex is differentiable a EIR few = a 121.24 axo X20 let = f(x) = a Fax 220 eve just need to see cheek differentiability at 2=0 Otherwise it is differentiable enerywhere. f'so) - lim floth)-fo), lim abs-o-tim a ht=0 tº h0 190 h - hao

f'(0) = heima f(0-1) = f (0) hto -h.. Elin – ans hao h 0 LH:D = > f(x) ice fea) R.H.D . is also differentiable at x = 0 enerywhere * QEIR. is differentiable ** - -40 -htoh 2* XTO f(x) = 8x2 = 20 L.H.D at x=0 f'col=hton f'ole im froth) – flol = lim ah -0 = -a hao h R.H:D at x = 0 f410= lim froth) - from lim 23 h : 0. – - Lim BK0 fex is differentiable oner IR 9 L.H.D - R.H.D at x=0 – –&=0 & aco] since fees is differentiable at 2=0 therefore it must be continuous at x=0. s tim fala lim fix) = fol XHOT - 100 =0 on B rre B is arbitrary. restriction no » There is rence sa=0) P[BEIR 20 at 2=2. # fex = lng f'(x)= . f(x)=2 = equation of tangent at x=2, 12) =