Find the intervals in which the function f(x) = x^3 - 12x^2 + 36x… - Vidyadip

Question

Find the intervals in which the function $f(x) = x^3 - 12x^2 + 36x +17$ is $(a)$ increasing, $(b)$ decreasing.

✓

Answer

$f'(x) = 3x^2 - 24x + 36 = 3(x^2 - 8x + 12) = 3(x - 2) (x - 6)$
$f'(x) = 0 \Rightarrow x = 2, x = 6$
$\therefore$ Possible intervals are $(-\infty, 2), (2, 6), (6, \infty)$
Since $f' (x) > 0$ in $(-\infty, 2), (6, \infty)$
$\therefore f(x)$ is increasing in $(-\infty, 2),\cup (6, \infty)$
And $f' (x) < 0$ in $(2, 6) \therefore f(x)$ is decreasing in $(2, 6).$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from APPLICATION OF DERIVATIVES→APPLICATION OF DERIVATIVES — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Draw a rough sketch of the curve $\text{y}=\frac{x}{\pi}+2\sin^2\text{x}$ and find the area between x-axis, the curve and the ordinates $\text{x}=0\text{ and }\text{x}=\pi.$

Find the area of the ragion in the first quadrant bounded by the parabola $y = 4x^2$ and the lines $x = 0, y = 1$ and $y = 4.$

$\text{If y}=3\cos(\log \text{x})+4\sin(\log\text{x}),\text{ show that }\text{x}^2\text{y}_2+\text{xy}_1+\text{y}=0$

Find the equation of the plane passing through the line of intersection of the planes $2x - y = 0$ and $3z - y= 0$ and perpendicular to the plane $4x + 5y - 3z = 8$.

Find the absolute maximum and the absolute minimum value of the following functions in the given intervals:
$\text{f}(\text{x})=(\text{x}-2)\sqrt{\text{x}-1}\ \text{in}\ [1,9]$

Evaluate the following intregals:
$\int\frac{\text{x}^2}{(\text{x}-1)(\text{x}+1)^2}\ \text{dx}$

Evaluate the following integrals:
$\int_{0}^\limits{1}\text{x}\log(1+2\text{x})\text{dx}$

Solve the following initial value problems:
$\frac{\text{dy}}{\text{dx}}-3\text{y}\cot\text{x}=\sin2\text{x},\text{ y}=2,\text{ when x}=\frac{\pi}{2}$

The top of a ladder 6 metres long is resting against a vertical wall on a level pavement, when the ladder begins to slide outwards. At the moment when the foot of the ladder is 4 metres from the wall, it is sliding away from the wall at the rate of 0.5m/ sec. How fast is the top-sliding downwards at this instance?
How far is the foot from the wall when it and the top are moving at the same rate?

Differentiate the following functions with respect to x:
$\sin^{-1}\Big(\frac{\text{x}}{\sqrt{\text{x}^2+\text{x}^2}}\Big)$