Find the local linear approximation of f(x) = e^3x at x = 1.

y = e³

y = e^{3(x - 1)}

y = 3e³(x - 1)

y = 3e³x - 2e³

Linearization of f(x) is its approximation using tangent line

y= f(xo) +(x-xo) f'(xo)

here xo=1

y= f(1) +(x-1) f'(1)

here xo =1

f(x)=e^3x

differentiate both sides w. r. t. x

f'(x) =d( e^3x ) /dx =3e^3x

now f(1)=e³

f'(1)=3e³

now putting values

y=e^3?+(x-1)3e³

^?y=e^3?+x3e^3?-3e³

y=3xe^3?-2e³

So last option is correct.