Find the maxima and minima, if any, for the function f given by
f (x) = x4 - 14x2 + 24x + 6.
Also find the values of the function at those points.
Given, f(x) = x4 – 14x2 – 24x + 6
f’(x) = 4x3 – 28x – 24,
f’’(x) = 12x2 – 28.
For maximum and minimum f’(x) = 0.
i.e. 4x3 – 28x – 24 = 0
or, x3 – 7x – 6 = 0.
Or, x3 – x2 – x2 – x – 6x – 6 = 0
Or, x2(x + 1) – x(x + 1) –
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If f (x) = 4x2 - 7x - 2/x - 2, x ≠ 2 find a ∂ > 0 such that |f (x) - 9| < 1/100 for 0 < |x - 2| < ∂. Hence show that lim x→2 f (x) = 9.
For which value(s) of k, is the function f, defined as below, continuous at x = 2?
Further, at which other points in [1, ∞ [ is f continuous, and why?
Trace the curve x = a (θ - sin θ ), y = a (1 - cos θ ). State the properties you use for tracing it, also.
Find the volume of the solid obtained by revolving the curve
X = a cos3 θ , y = a sin 3 θ about the y-axis.
A jogger runs from her home to a point A, which is 6 km away. For there 6 km, she begins by running at a constant speed till she reaches a hilly portion 2 km from her home. Here her speed slows down while she runs up the hill, which is a 1-km run. Then she speeds up while running down the hill. The last 2 km of the run are again at
constant speed. Draw a graph to show the jogger’s speed as a function of the distance from her home. Also find the range of this function.
- hich of the following statements are true or false? Give reasons for your answers.
- The greatest integer function is continuous on R .
- The domain of the function f , given by
iii) The function f , defined by f (x) = x 3 - 6x 2 + 16x - 15, is increasing in R .
iv) Rolle’s theorem is applicable for the function f , defined by
f (x) = 1 + x 2 /3 in the interval [-1,1].v)
Calculate the approximate value of 
x 3 dx by taking 6 subintervals of equal length and applying Simpson’s rule.