Maxima And Minima

Step 1: Concept
Find and set it to zero for critical points.
Step 2: Analysis
.
.
Step 3: Conclusion
At , changes from positive to negative, confirming a maximum.
Final Answer: (B)

Step 1: Differentiation
Let . .
Step 2: Critical Points
Put .
Step 3: Verification
The second derivative . At , , which indicates a maximum.
Step 4: Calculation
Maximum value at is .
Final Answer: (d)

Step 1: Find Critical Points
. Set .
Step 2: Analyze Roots of
For extreme values to exist, must have real roots. Discriminant .
Step 3: Check Given Condition
Given . This means , so has no distinct real roots. If never changes sign, the function is monotonic and has no maximum or minimum values.
Final Answer: (c)

Step 1: Concept
Slope is given by . To maximize the slope, find the critical points of by setting .
Step 2: Analysis
. .
Step 3: Evaluation
Set in the given interval.
Step 4: Conclusion
Check at : . At , . Thus, slope is maximum at .
Final Answer: (b)

Concept: Use differentiation to find minimum value.

Step 1: Let .

Step 2: Set derivative to zero:

Step 3: Minimum value: Final Answer:

Concept: Minimize distance from origin:

Step 1: Use given constraint.


Step 2: Apply substitution or Lagrange multipliers.
Minimize: subject to constraint.

Step 3: Use symmetry approach.
Let:

Step 4: Solve relation.


Step 5: Substitute back.


Step 6: Compute minimum distance.
Conclusion:
Not in options None of these

Concept: Minimum occurs when expression becomes smallest possible value.

Step 1: Analyze components.


Step 2: Minimum possible value.


Step 3: Check when .
Occurs when:

Step 4: Check solution.
At : For , equation has no real solution due to dominance of polynomial.

Step 5: Conclusion.
Minimum occurs only at Conclusion:
Number of points = 1

Concept: Check left-hand limit, right-hand limit, and value at .

Step 1: Left-hand limit as .

Step 2: Value at .

Step 3: Right-hand limit as . Function is continuous at .

Step 4: Check derivative sign. For , near . For , . Thus, increases up to and then decreases → local maximum at .

Concept: Use differentiation to find maxima.

Step 1: Define the function.

Step 2: Differentiate.

Step 3: Set . Using : Let : (not in given interval)

Step 4: Check interval. lies in and is valid.

Step 5: Compare with options.

Concept: Use differentiation to find point of minimum for .

Step 1:Define the function:

Step 2:Differentiate with respect to :

Step 3:Set for critical point:

Step 4:Since :

Step 5:Verify minima using second derivative: Hence, it is a point of minimum.