Find the first derivative.

By the Sum Rule, the derivative of with respect to is .

Evaluate .

Since is constant with respect to , the derivative of with respect to is .

The derivative of with respect to is .

Evaluate .

Since is constant with respect to , the derivative of with respect to is .

The derivative of with respect to is .

Multiply by .

Find the second derivative.

By the Sum Rule, the derivative of with respect to is .

Evaluate .

Since is constant with respect to , the derivative of with respect to is .

The derivative of with respect to is .

Multiply by .

Evaluate .

Since is constant with respect to , the derivative of with respect to is .

The derivative of with respect to is .

The second derivative of with respect to is .

Set the second derivative equal to .

Divide each term in the equation by .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Move the negative in front of the fraction.

Replace with an equivalent expression in the numerator.

Remove parentheses.

Apply the distributive property.

Rewrite in terms of sines and cosines.

Apply the distributive property.

Multiply .

Combine and .

Combine and .

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Move the negative in front of the fraction.

Simplify each term.

Separate fractions.

Convert from to .

Divide by .

Multiply by .

Separate fractions.

Convert from to .

Divide by .

Multiply by .

Add to both sides of the equation.

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Take the inverse tangent of both sides of the equation to extract from inside the tangent.

The exact value of is .

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.

Simplify the expression to find the second solution.

Simplify .

To write as a fraction with a common denominator, multiply by .

Combine fractions.

Combine and .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Subtract from .

Move the negative in front of the fraction.

Add to .

The resulting angle of is positive and coterminal with .

Find the period of .

The period of the function can be calculated using .

Replace with in the formula for period.

The absolute value is the distance between a number and zero. The distance between and is .

Divide by .

Add to every negative angle to get positive angles.

Add to to find the positive angle.

To write as a fraction with a common denominator, multiply by .

Combine fractions.

Combine and .

Combine the numerators over the common denominator.

Simplify the numerator.

Move to the left of .

Subtract from .

List the new angles.

The period of the function is so values will repeat every radians in both directions.

, for any integer

, for any integer

Substitute in to find the value of .

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

The exact value of is .

Combine and .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

The exact value of is .

Multiply .

Multiply by .

Combine and .

Move the negative in front of the fraction.

Simplify by adding terms.

Combine the numerators over the common denominator.

Subtract from .

Divide by .

The final answer is .

The point found by substituting in is . This point can be an inflection point.

Substitute in to find the value of .

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

The exact value of is .

Combine and .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

The exact value of is .

Multiply .

Multiply by .

Combine and .

Move the negative in front of the fraction.

Simplify by adding terms.

Combine the numerators over the common denominator.

Subtract from .

Divide by .

The final answer is .

The point found by substituting in is . This point can be an inflection point.

Determine the points that could be inflection points.

Split into intervals around the points that could potentially be inflection points.

Replace the variable with in the expression.

The final answer is .

At , the second derivative is . Since this is negative, the second derivative is decreasing on the interval

Decreasing on since

Decreasing on since

Replace the variable with in the expression.

The final answer is .

At , the second derivative is . Since this is positive, the second derivative is increasing on the interval .

Increasing on since

Increasing on since

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are .

Find the Inflection Points f(x)=7sin(x)+7cos(x)