MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
The function g(x)=∣x∣ is defined for all real numbers. Which statement is true regarding g′(0), the derivative of g at x=0?
(a) g′(0)=1 because, for small positive values of x, g(x)=x, so the derivative there is 1 and extends to x=0.},{
(b) g′(0) does not exist because the slopes of the secant lines approaching 0 from the left and right are different.
(c) g′(0)=0 since the function attains an absolute minimum at x=0, so the tangent line must be horizontal there.
(d) g′(0) does not exist because the denominator of the difference quotient is 0, even though the slopes from both sides are the same.