Find the maximum value of + . - Vidyadip

Question

Find the maximum value of $\sin \theta+\cos \theta$.

✓

Answer

Given $\quad f(\theta)=\sin \theta+\cos \theta, \theta \in[0, \pi]$
$\therefore \quad f^{\prime}(\theta)=\cos \theta-\sin \theta$
$f^{\prime}(\theta)=0$
$\Rightarrow \quad \sin \theta-\cos \theta=0 \quad \Rightarrow \tan \theta=1$
$\therefore \quad \theta=\frac{\pi}{4}$
$\begin{array}{l}
\text { hence } f(0)=\sin 0+\cos 0=1 \\
f\left(\frac{\pi}{4}\right)=\sin \frac{\pi}{4}+\cos \frac{\pi}{4}=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\sqrt{2} \\
f(\pi)=\sin \pi+\cos \pi=0-1=-1 \\
\therefore \text { Absolute maximum value }=\sqrt{2}
\end{array}$

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 "3 Marks" from Applications of Derivative→Applications of Derivative — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Differentiate the following w.r.t. x:
$\sec^{-1}\Big(\frac{1}{4\text{x}^3-3\text{x}}\Big),0<\text{x}<\frac{1}{\sqrt{2}}$

Show that $\text{y}=\text{ax}^3+\text{bx}^2+\text{c}$ is a solution of the differential equation $\frac{\text{d}^3\text{y}}{\text{dx}^3}=6\text{a}$

If $\text{y}=\log\Big(\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}}\Big),$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\text{x}-1}{2\text{x}(\text{x}+1)}$

A random variable X has the following probability distribution:

Values of X:	0	1	2	3	4	5	6	7	8
P(X)	a	3a	5a	7a	9a	11a	13a	15a	17a

Determine:

$\text{P}(\text{X}<3),\text{P}(\text{X}\geq3),\text{P}(0<\text{X}<5).$

Prove that the Greatest Integer Function f: R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

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

Using vector method, prove that the point is collinear:
A(2, -1, 3), B(4, 3, 1) and C(3, 1, 2)

Form the differential equation of the family of curves y = a cos(x + b), where a and b are arbitrary constants.

Write the following in the simplest form:
$\tan^{-1}\Big\{\sqrt{1+\text{x}^2}-\text{x}\Big\},\text{x}\in\text{R}$

Prove that the function f : R → R defined by f (x) = 2x + 5 is one-one.