For any twice differentiable function 𝑓: ℝ → ℝ, if at some 𝑥∗ ∈ ℝ, 𝑓′ (𝑥∗ ) = 0 and 𝑓′′(𝑥∗ ) 0, then…

Question

For any twice differentiable function 𝑓: ℝ → ℝ, if at some 𝑥∗ ∈ ℝ, 𝑓′ (𝑥∗ ) = 0 and 𝑓′′(𝑥∗ ) > 0, then the function 𝑓 necessarily has a ______ at 𝑥 = 𝑥∗ .

Note: ℝ denotes the set of real numbers.

Accepted Answer

Correct answer: A — Key idea: use the second derivative test. Since f′(x*) = 0, x* is a critical point. Since f′′(x*) > 0, the graph is concave up near x*. Using the local Taylor expansion, f(x) ≈ f(x*) + (1/2)f′′(x*)(x − x*)² near x*. The quadratic term is positive for x close to x* and x ≠ x*, so nearby values of f(x) are greater than f(x*). Therefore, f has a local minimum at x = x*. This condition does not by itself guarantee a global minimum.

Answer

A. local minimum

Answer

B. global minimum

Answer

C. local maximum

Answer

D. global maximum

For any twice differentiable function 𝑓: ℝ → ℝ, if at some 𝑥∗ ∈ ℝ, 𝑓′ (𝑥∗…