A particle moves along a line according to the law s(t) = 2t³ − 9t² +12t − 4, where t ≥ 0.

A particle moves along a line according to the law s(t) = 2t³ − 9t² +12t − 4, where t ≥ 0.

A police jeep, approaching an orthogonal intersection from the northern direction, is chasing a speeding car that has turned and moving straight east. When the jeep is 0.6 km north of the intersection and the car is 0.8 km to the east. The police determine with a radar that the distance between them and the car is increasing at 20 km/hr. If the jeep is moving at 60 km/hr at the instant of measurement, what is the speed of the car?

Find the slope of the tangent to the following curves at the respective given points
y = x⁴ + 2x² − x at x = 1

Find the equations of the tangents to the curve y = 1 + x³ for which the tangent is orthogonal with the line x +12y = 12.

Find the equations of the tangents to the curve y = \(\frac{x+1}{x-1}\) which are parallel to the line x + 2y = 6.

Find the intervals of monotonicities and hence find the local extremum for the following function:
\(f(x)=\frac { x }{ x-5 } \)

Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals.
\(f(x)=|\frac{1}{x}|, x\in [-1,1]\)

Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions:
\(f(x)=\frac{x^{2}-2x}{x+2}, x\in [-1,6]\)

Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions:
\(f(x)=\sqrt{x}-\frac{x}{3}, x\in [0,9]\)

Show that the value in the conclusion of the mean value theorem for
\(f(x)=\frac{1}{x}\) on a closed interval of positive numbers [a, b] is \(\sqrt{ab} \)

Find intervals of concavity and points of inflexion for the following function:
f(x) = sin x + cos x, 0 < x < 2

Does there exist a differentiable function f(x) such that f(0) = -1, f(2) = 4 and f'(x) ≤ 2 for all x. Justify you answer.

Find two positive numbers whose product is 20 and their sum is minimum.

Find the smallest possible value x²+y² given that x +y = 10.