Applications of Differentiation EXERCISE 5
1. Find the stationary point(s) of the functions below and state if these are local maxima or local minima:
(a) y = x 2 − 4x + 5, x ∈ R (b) y = − 3x 2 + 18x − 5, x ∈ R (c) y = x 3 − 3x 2 − 9x − 3, x ∈ R (d) y = 11 − 12x + 9x 2 − 2x 3 , x ∈ R (e) y = − x 3 − 3x 2 + 7, x ∈ R
2. Find the maximum value of y = − 2x 2 − 8x + 7, x ∈ R.
3. Find the stationary point of y = ax 2 + bx + c and show it is a local maximum if a < 0 and a local minimum if a > 0.
4. The graph of a cubic function y = f (x) in the domain −2 ≤ x ≤ 5, x ∈ R, is shown.
y (–2, 4) –2 –1
(i) the local minimum (ii) the local maximum
(b) Find:
(i) the maximum value of y in the domain (ii) the minimum value of y in the domain
5. Find the local maximum and local minimum of y = 4x 3 − 6x 2 − 24x − 14, x ∈ R.
The graph of y = 4x 3 − 6x 2 − 24x − 14, −2 ≤ x ≤ 3, x ∈ R, is shown. y
B –2 –1 A D 317 C 0 12 3 E x (3, 5) 5 x (a) Write down the co-ordinates of:
(a) Find the co-ordinates of A, B, C, D and E. (b) What name is given to B? (c) What name is given to D?
(d) Find the range of values of x for which y is decreasing.
(e) What is the maximum value of y in the domain?
(f) What is the minimum value of y in the domain?
6. (a) If f (x) = x 2 + px + 10, x ∈ R, fi nd f ʹ(x). If the minimum value of f (x) is at x = 3, fi nd p ∈ Z.
(b) If f (x) = 3 + 8x − 2x 2 , x ∈ R, fi nd f ʹ(x).
(i) Find where the curve crosses the y-axis.
(ii) Find the maximum value of f (x). (iii) For what range of values is f ʹ(x) > 4?
(c) If f (x) = x 3 − 3x 2 + k x + 2, x ∈ R has a stationary point at x = 1, fi nd k and the co-ordinates of the stationary point.
(d) The slope of the tangent to the curve y = x 3 − k x + 7, x ∈ R at x = 1 is −9, fi nd k ∈ R. Hence, fi nd the co-ordinates of the local maximum and local minimum of this curve.
(e) f (x) = (x + k)(x − 2 ) 2 . If f (3) = 7, fi nd k ∈ R. Hence, fi nd the local maximum and local minimum of f (x).
(f) f (x) = ax 3 + bx + c. Find a, b, c, ∈ R, if:
(i) f (0) = 3 (ii) the slope of the tangent at x = 1 is –18 (iii) the curve has a local maximum at x = 2
(g) If f (x) = x 3 − 3x 2 + k x + 1 has a stationary point at x = −1, fi nd k ∈ R. Find the local maximum and local minimum of f (x).
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