If $ f(x)=\left\{\begin{array}{cc}3^{x}, & -1 \leq x \leq 1 \ 4-x, & 1

Question

If $ f(x)=\left\{\begin{array}{cc}3^{x}, & -1 \leq x \leq 1 \ 4-x, & 1 < x < 4\end{array}\right. $, then at $ x=1, f(x) $ will be, Continuous but not differentiable, Neither continuous nor differentiable, Continuous and differentiable, Differentiable but not continuous

MARKS App · Accepted Answer

Since f 1 - 0 = lim x → 1 3 x = 3 f 1 + 0 = lim x → 1 4 - x = 3 and f(1) = 3 1 = 3 ∴ f 1 - 0 = f 1 + 0 = f 1 ⇒ f x is continuous at x = 1 Again f ′ 1 + 0 = lim x → 1 + f x - f 1 x - 1 = lim x → 1 3 x - 3 x - 1 = lim h → 0 3 1 + h - 3 h = 3 lim h → 0 3 h - 1 h = 3 log 3 and f ′ 1 + 0 = lim x → 1 f x - f 1 x - 1 = lim x → 1 4 - x - 3 x - 1 = - 1 ∴ f ′ 1 + 0 ≠ f ′ 1 - 0 ⇒ f(x) is not differentiable at x = 1

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