GATE DA Calculus and Optimization Quiz

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.

Consider the function 𝑓: ℝ → ℝ where ℝ is the set of all real numbers.

f(x)= (x4/4) - (2x3/3) - (3x2/2) +1 

Which of the following statements is/are TRUE ?

Evaluate the following limit:

lim _𝑥→0 ln((𝑥 ²+1) cos 𝑥)/ 𝑥 ² = ______

Let 𝑓:ℝ → ℝ be the function 𝑓(𝑥) = 1 /1+𝑒^−𝑥 . The value of the derivative of 𝑓 at 𝑥 where 𝑓(𝑥) = 0.4 is ______ (rounded off to two decimal places).

Note: ℝ denotes the set of real numbers

Let 𝑓:ℝ → ℝ be a function. Note: ℝ denotes the set of real numbers.[Tex]f(x) = \begin{cases} -x, & \text{if } x < -2 \\ ax^2 + bx + c, & \text{if } x \in [-2, 2] \\ x, & \text{if } x > 2\end{cases}[/Tex]

Which ONE of the following choices gives the values of 𝑎, 𝑏, 𝑐 that make the function 𝑓 continuous and differentiable?

Let [Tex]f(x)=\frac{e^{x}-e^{-x}}{2},x \in \mathbb{R}.[/Tex] Let f ^(k) (a) denote the k^th derivative of f evaluated at a. What is the value of f ⁽¹⁰⁾(0)? (Note: ! denotes factorial)

Consider two functions f : R → R and g : R → (1, ∞). Both functions are differentiable at a point c. Which of the following functions is/are NOT differentiable at c? The symbol · denotes product and the symbol ◦ denotes composition of functions.

[Tex]\lim_{t \to +\infty} \left( \sqrt{t^2 + t} - t \right)[/Tex]

Consider the function f(x) = x³/3 + 7/2x² + 10x + 133/2 , x ∈ [−8, 0]. Which of the following statements is/are correct? 

Let f : R → R be a twice-differentiable function and suppose its second derivative satisfies f''(x) > 0 for all x ∈ R. Which of the following statements is/are ALWAYS correct?