241.

Find the local maxima or local minima, if any, of the following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
$straight h left parenthesis straight x right parenthesis space equals space sinx plus cosx comma space 0 space less than space straight x space less than space straight pi over 2$

75 Views

242.

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
$sinx minus cosx comma space space 0 less than straight x less than 2 straight pi$

72 Views

243.
Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
f(x) = x³ – 6x² + 9x + 15

Name: Zigya - For The Curious Learner
Rating: 4.4 (740 reviews)

Let f (x) = x³ – 6x² + 9x + 15
∴ f ' (x) = 3x²- 12x + 9 = 3 (x² – 4 x + 3) = 3 (x – 1) (x – 3)
f ' (x) = 0 ⇒ 3 (x – 1) (x – 3) = 0 ⇒ x = 1, 3
When x < 1 slightly, f ' (r) = 3 (– 1)(–) = + ve
When x > 1 slightly, f ' (x) = 3 (+)(–) = – ve
∴ at x = 1, f ' (x) changes from + ve to – ve
∴ f (x) has local maxima at x = 1
and local maximum value = 1 – 6 + 9 + 15 = 19
When x < 3 slightly, f ' (x) = 3 (+) (–) = – ve
When x > 3 slightly, f ' (x) = 3 (+) (+) = + ve
∴ at x = 3, f ' (x) changcs from – ve to + ve
∴ f ' (x) has local minimum value at = 3
and local minimum value = (3)³ – 6 (3)² + 9 (3) + 15 = 27 – 54 + 27 + 15 = 15

101 Views

244.

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
f(x) = (x – 1)(x + 2)²

81 Views

245.

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
$straight g left parenthesis straight x right parenthesis space equals straight x over 2 plus 2 over straight x comma space space space straight x greater than 0$

76 Views

246.

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
$straight g left parenthesis straight x right parenthesis space equals space fraction numerator 1 over denominator straight x squared plus 2 end fraction space equals space left parenthesis straight x squared plus 2 right parenthesis to the power of negative 1 end exponent$

71 Views

247.

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
$straight f left parenthesis straight x right parenthesis space equals space straight x square root of 1 minus straight x end root comma space 0 less than straight x less than 1$

74 Views

248.

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
$sin to the power of 4 straight x plus cos to the power of 4 straight x comma space space space 0 less than straight x less than straight pi over 2$

73 Views

249.

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
$sin 2 straight x minus straight x comma space space minus straight pi over 2 less or equal than straight x less or equal than straight pi over 2$

78 Views

250.

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
$left parenthesis straight x minus 3 right parenthesis to the power of 4$