Calculus Answers

Let f : [0, 4] → R be a thrice differentiable function in (0, 4) such that

f(0) = −1, f(1) = 2, f(3) = −2, f(4) = 4.

Let g(x) = f(x)f '(x)f''(x) + 4. Using the mean value theorem, find the minimum number of distinct roots of g'(x) = 0.

State true or false giving proper justification for following statement:there exists a continuous function f:RtoR2 such that f(down)=(n,1/n) for all n belongs to N

Trace the curve y

2 = (x − 1)(x − 2)(x − 3).

Find the Tangent and Normal line to the given curve. 9x^3 - y^3 = 1 at (1,2)

Find the Tangent and Normal line to the given curve. y=√(16+x^2) at the origin

Values of f(x) are given at a, b, and c. Show that the maximum is obtained by

f(a) (b^2 - c^2) + f(b) (c^2 - a^2) + f(c) (a^2 - b^2)

x = -----------------------------------------------------------------

f(a) (b - c) + f(b) (c - a) + f(c) (a - b)

"\\int" Cosec⁶(2x) dx

Given that [𝑥] denotes the greatest integer function

a) Sketch the graph of 𝑓(𝑥)=[𝑥]−𝑥2 𝑓𝑜𝑟 𝑥∈[0,4]

Trace the curve x3 + y2 = 2axy